I’m having another go at playing my drow monk, and I’m trying really hard to start her off with the typical drow mentality so that she can get some natural character development by being with the party.

One of the most important aspects of drow culture is that it’s considered the pinnacle of weakness and shame to ask for help. But what does “asking for help” even mean? Is it only in the most literal sense, or does it include things like accepting help from others without asking them outright? And what defines “help” anyway? Most drow are slave owners, so apparently what they order their slaves to do doesn’t count. What about commanding a service from someone who isn’t a slave? What about paying for goods and services? Where is the line between asking and ordering?

In an adventuring party, being able to ask for help is pretty much a requirement, and of course Xaira (my drow) will have to learn that. But she’s still fairly new to the party. I’ve been going with the notion that if she needs something from the other party members, she’ll try to make it as clear as possible that she wants and expects it from them without literally asking for it. Would you do the same, or would you play it some other way?

It does matter why you're with the party now, because the pieces of Drow culture you've rejected will inform your roleplay.  That said...

As a female Drow, you're presumably accustomed to being in power over male Drow, who themselves are in power over non-Drow (the slaves you mentioned).  The former meme is more deeply entrenched in Drow culture - sexism matters more than speciesism, if you have to pick one, i.e. a female Drow probably thinks a man in power is far more bizarre than a human in power.  As you correctly point out, "asking" (telling) males and non-Drow for "help" isn't weakness, and I wouldn't play it as such. 

So especially if there are no other races in the party I'd have a particular issue with, like a fellow Drow, or a half-Drow, or what have you, I'd basically act like I was simply entitled to the other party members helping me.  There'd be a noticeable lack of the word "please" coming out of my mouth, and I'd tell, not ask, when I wanted something done.  Never "Would you grab me that pitcher?", always "Grab me that pitcher!", etc.  Assuming the party had at least one other female in it, I could play it two ways - stereotypically, I'd view the other female as a rival, in much the same way a spider will eat another spider given half a chance.  But that's very toxic to a practical party dynamic, so I'd probably replace it with trying to learn from my fellow female how to succeed in the surface world.  So I'd make her my role-model, and look up to her, by default.

I think Quinn’s advice is solid. Also, the tension between detesting asking for help (especially from males), but being put in surface world situations where it’s needed for survival, that sounds like a good RP and growth opportunity. “Tough kid learning to drop the act through surviving hardship with companions who become friends” is a common enough trope in literature of all sorts, you’ll find some inspiration out there. But if you’re going to start out unfriendly/sexist/racist/evil, the important parts are (1) finding reasons to keep yourself with the group, that aren’t satisfied by a single quest, and (2) balancing party rivalries with party bonds. Sneering at the boys is fine, as long as you cozy up to at least one ally (even if it’s an act), because you can’t just be the party member that hates everyone and nobody likes. 

This is where some some positive metagaming can come in handy. All of the above advice is very good, but it's a tough acting challenge to portray that mix of arrogance and hidden vulnerability that you're going to want to pull off the "Tough on the outside, but soft on the inside" kind of character so I would fully suggest that you make out of character comments about your character's emotions to the other players so they know very clearly the kind of character arc you're going for and can play along rather than have friction at the player level. Oh and definitely bring it up in Session 0. I find that tricky character arcs like this go much better if you get the cooperation and buy-in form the other players at an explicit level.

Never "Would you grab me that pitcher?", always "Grab me that pitcher!" - You're never going to find True Love if you're not polite.  :-)

Playing a "nice" Drow, running around on the surface world, and joining a bunch of Adventurers is already completely breaking the stereotypical behavior of Drow.  Sure, you can lean into the stereotype and be rude to people in your roleplay, but that borders on what they call an "Edgelord", a character with a dark background who only cooperates begrudgingly with people who are trying to be their allies.  Where's the fun in that?

I'd suggest that instead, you play your character as being extremely polite.  Don't overdo it, be subtle most of the time, but if your character needs help with something, ask freely, and with as much courtesy as you can pile on.  "Westly, I don't want to bother you, and I know that the pitcher is right there in front of me, but if you would be so kind, would you please fetch me that pitcher?"

Save your character's hostility for the bad guys.  It's ok to be rude to them.

For context, I do have reasons why she's on the surface and joined the party. The backstory I gave her has her family maliciously abandon her on the surface, and the DM introduced her to the party by having them free her from a trap that she fell into in a dungeon.

I'm seeing a consensus that demanding help isn't the same as asking for it. That makes sense, to an extent. I do like the suggestion of bonding with the other woman in the party. She was the first one to impress my character in combat, so it seems natural.

I currently play a drow female and having a lot of fun with it. For RP, it is best to remember you are a stranger n a strange world. Best is to model her behavior as you might have experienced a foreign country or your impressions of foreigners in your country.  Speak Common in an accent and ask why is this thing a certain way. You can be the foreigner in love with her new surroundings and trying to adapt to local custom or you can play as the "Ugly American" where you are upset at everything not being the same as in the Underdark. Complain about the sun being too bright, when the group decides to ride horses, ask if they have a riding lizard instead, etc.

As far as the male thing, suggest no overt fighting, but you can get in digs when you can. For example, a male character makes a decision that works out badly, mutter something about tiny male brains. Another fun RP is to play on what she would have been taught about other species. For example, we have a bugbear character and initially she would say things like "hmm, you are pretty bright for a bugbear." Of course, the more she gets to know the party, the more those type of tropes or passive:aggressive behaviors should fade.

I have been running Drow campaigns for over 10 years. They are my favorite group of people tp play, Elves who have been Selectively Breeding for 30,000 years to be smarter, more cunning, more adept, more attractive, with Agency, and Ambition. They are all the things Corellon kept Elves from being, Masters! AD&D started my love.

Female Drow, Evil ones, do not look at other women as a Role Model,. Always as a Rival but Respect will be shown 

As has been pointed out Drow simply do not ask for help. From the Lesser Races, i.e. all non-Drow, it is simply implied. As a female Drow, it is to expected. Drow women are raised from birth to be Superior. From a man of any species In the Olden Days of D&D, there was a huge difference between Female and Male Drow stat wise..

Now the important part is alignment. If she is Good, then all advice above its perfect! If she is Evil, things get a little more fun. If she is Evil then Morality makes a huge difference as well. Jaraxle and Brizza, totally different outlooks.

Chaotic Evil , depending on how it is played doesn't mean "I am burning the the entire village down because!" Drow are Elves, they are not stupid, ignorant on surface things yes, but not on how the Power Dynamitic works. She would catch on quickly and modify and temper her words and actions to keep herself from getting murdered, since she has no back-up. She would also be open to "befriend" the surface folk to help take revenge on her House. Chaotic Drow can be hypocritical. They live in organized cities with Laws and Rules. They do not destroy the environment for the sake of destroying. They are Vegan, as per the 2nd and 3rd Editions of Drow of the Underdark, yet have Zero Value on humanoid life.

If she is Lawful then things become a lot more smooth to navigate. Should be easier to adapt to local customs and quick to figure out local laws. Also easier to be more open to make friends in the group if they themselves are Lawful. She will still be able to work inside the group and be able to take advantage of loopholes.

The biggest character flaws in the Drow are their Ego and Hubris. That is something that can be difficult to overcome. Drizzt even has issues with that. It also makes them one of the most complex people out there.

Sorry for my rambling. The Drow are near and dear to heart because they are the best Elf an Elf can be mentally and physically and total hypocrites! 

Wow, no need to apologize. I love it.

In terms of alignment, I'm trying to play her like she was raised to be Chaotic Evil but being on the surface has forced her towards Chaotic Neutral. I should probably play up her ego more - the only power trip I've given her is when she agreed with an NPC that she was the leader of the party, even though she had just joined.

I misunderstood what you were trying to play.  How would a Chaotic Neutral Drow on the surface world behave towards a party of adventurers she joined?  Naturally, they should all feel honored that you chose to join them and that goes double for any female members of the party related in any way to Drow.  It should go triple for any males,  and quadruple for male Drow!.   Chaotic Neutral's only care about themselves and try to get others to feel the same way.  If you're feeling happy, I'd say you would sit there and bask silently, marveling in your own magnificence.  Talking to anyone of equal or lesser station would just be irritating.  Remember, you're a member of the only true "Master Race."

If you're not happy, obviously, you should make everyone else as miserable as you can.  You probably would want to attack them, but you know how Drow behave, they're almost the same as you, just nastier.  I'd say you'd hold off on attacking and just be as rude as you could possibly be.  Bump people into the campfire.  Belittle their every effort to do anything.  If you want a pitcher, you don't ask, you just go up and grab it, and then yell at them for being in your way.

Given that you're sensitive to sunlight, and hindered in the light of the day, and at night people are going to want to do foolish things like sleep...  You don't ever really need to sleep!  You've got Trance!  Or they might waste time talking to each-other when they should be worshipping the ground you walk on, and like all Drow, you're the victim of the stereotypical Drow elf behavior...  They're pretty nasty folks.  They abandoned you instead of letting you seek your rightful destiny after all...  It's unlikely that you'd ever really be happy.

You asked about how a Drow might go about asking for help.  I'd say that would be "never".  Drow issue commands, they demand to be served, and if they don't get what they want...  they set something up so their inferiors get killed.  That's how Drow politics works.

I'm sure you'll have a lot of fun, no matter whose advice you take.

Drow may have cultural norms about matriarchy, but that isn’t to say they don’t have individual personalities that run the spectrum. I’m sure there are shy drow, drow that lack social confidence and fall in line with stronger personalities, easygoing drow, rambunctious or goofy drow, etc. Heck, maybe they’re starting from a drow culture perspective, but are drawn to stay on the surface because there are elements of it that better fit their personality. Or maybe they DO like drow culture, but find that foreigners give them more respect/fear than they get at home in competitive drow hierarchy. That’s not even getting into alignment….

Lots of ways to intersect background, present, and future. There’s no “right way” to do this, so don’t back yourself into “I have to be a jerk to the party or torpedo the quest, it’s what my character would do” territory with drow stereotypes.

What you did was Totally Drow!

The Drow Ego is Legendary. LOL Drizzt, who is Chaotic Good has shown it many times. Jaraxle doesn't hide his Ego one bit, even to Matron Mothers. Both of which are male Drow not even near the level of Ego the girls have .LOL I can see her alignment to Neutral is a really good way to adapt. The Surface World would be a Culture Shock.

I have a Drow Assassin from the old 3.5 Drow campaign brought over to 5th. She was Chaotic Evil but she switched to Chaotic Neutral as well. She is totally Drow Ego. She hates the "Lesser Races" and hates seeing a woman of any race not in a place of authority. She totally loathes Sun and Moon Elves but  will get involved if she sees a woman of either race abused. She will kill them later but the point is she grew up in a Matriarchy Society. Girls Rule Boys Serve. That is the way she is and she has Zero Tolerance for the Lesser Races on the Surface that do not share the same view point. She is also quite Overpowered with her gear and stats. I use her to remind my group about how Drow are. LOL

Totally digressed again!

How is your Drow turning out?

Was that directed at anyone in particular?