Hey everybody. So I'm starting a new campaign with six players, and all of us have a fair amount of experience. Right off the bat it sounds fun, because in my head the party is balanced.
Goliath Totem Barbarian.
Wood Half-elf Arcane Trickster Rogue.
Firbolg Flame (Homebrew) Druid.
Shadar-Kai Undying Warlock.
Mark of Passage Human Shadow Monk.
Variant Human Glory Paladin.

I've played with these people before and their playstyle is... unique. Not bad, mind you, just different. I was wondering if I could post issues we have here and get advice on how to handle things. Any input is welcome.

The first issue, or rather thing I see becoming an issue, is the Undying Warlock wants to play as an observer. They want to stand around taking notes on things while the others fight and talk and stuff. How should I address this, and does it even need addressing?

Have a talk with the warlock and remind them D&D isn’t like a video game where you can do whatever you want. You need to work together and make sure everyone else is having fun. He can play a researcher, but he also needs to fight alongside his companions when the time comes. If he wants to be a party member out of convenience while taking notes in the field, that’s fine, but he needs to back up the party when they’re in danger, not just because it’s what any reasonable person would do, but because it’s a cooperative game.

I agree with Naivara as usual. You need to have a conversation with the player and bring up the points mentioned.

A player who creates a "background" character probably falls under one or more categories:

1) They don't actually want to play.

It's possible that they only really care about spending time with the group and aren't actually invested in the game.
It's possible that there is someone, or multiple someones, in the group that they are having issues with. They are physically present, but are too distracted to get immersed.

2) They are introverted and need encouragement.

Sometimes being an "observer" is a defense mechanism and those players can benefit from being directly engaged. Have the NPCs talk to the players individually, rather than as a group. Ask them, individually, how they respond to an interaction, or what their character is doing while another character has the spotlight.

3) They aren't invested in their character or the storyline.

It's easy to get excited about an idea for a character and then not know what to do with it in an actual game. Race and Class inform "what" we can do, but don't answer the "how" and "why". Some players need to be woven into the world to know their place within it. This can be encouraged by having one-on-one sessions for character development, and by establishing relationships between the players and NPCs.

If two players have a common secret, goal, or patron, it becomes easier for them to lean on one another when there isn't an obvious path forward. Their "partner" becomes a tangible motivator that gives their character purpose.

If a player is hiding from adventure, let the adventure find the player. Warlocks are particularly good for this because they have a supernatural patron who is invested in their journey.

When the player is writing notes in a tavern, have the book pulse with arcane energy and slam on the table as the barmaid is walking by. The barmaid can either respond flirtily, as though the player were trying to get her attention, or angrily for essentially the same reason. When the Warlock resolves the encounter, he can open the book to find someone else's handwriting, a message from his patron.

A Warlock's patron wants something, and an "observer" isn't getting that done, whatever it is. Use that to push the Warlock into situations that can't be ignored without consequence.

Thank you all for the very good points. I'll go over these with the player and try to implement some of these with the patron. Be on the look out for more advice giving moments like this. :)

It may be instructive to find out why they think that they only want to be an observer all the time.

Do they just really not want to play (they are an experienced Player, but maybe this Campaign just doesn't grab them yet)? Are they an "audience member" style Player - to quote Matt Colville - and they're really there to watch the story unfold? Are they shy, and have struck upon this as a means of not needing to put themselves out there, since they're "just [not] doing what my Character [wouldn't] do"?

They're an experienced Player, so not all of these are likely - but you should understand what their approach to your Campaign actually is.

And I'm not completely convinced this a) is an issue yet ( since you phrased this as "wants to", not "is doing" ), and b) isn't something that would sort itself out in play, or through Party interactions, without you needing to intervene. Are they really not going to to jump into the conversation when something they want is on the line, and no one else is bringing it up? Are they really going to just watch and take notes as the Ogre breaks off from the main combat and charges them? And if this is something they actually ( don't ) do, do you not think that the other Players might have something to say about that? 

You may be borrowing potential trouble from the future, and you might not need to intervene at all.

However, If this is something you end up needing to address, you do need to know the Player's motivations for acting that way. I don't think you can try and mold someone's playing style into something that you personally would like, browbeating someone with their Player duty is likely to cause resentment, and one of strengths of TTRPGs over video games is the fact that you can do whatever you like - but the approaches and playing styles of everyone around the table has at least be compatible.

Talk with the player and figure out what the character wants and work that into the story as his/her hook.  There has to be a reason why a character leaves the safety of home for the hazardous and lethal great beyond.  What is the purpose of observing?  Is he trying to catalog every type of monstrosity to write a book or an academic paper to earn recognition back home?  Is she trying to track down and identify artifacts from a lost civilization to use them to save her own society? 

A warlock has a patron, and that right there is a fantastic hook built into the character from the beginning.   How did the pact come into existence?  Has the patron made some sort of demand or have an ulterior motive in exchange for granting warlock powers to the character?  Maybe the patron demands action and great deeds of the character in direct opposition to the "I just want to stay back and make notes" attitude?  That could create good roleplaying tension between what the character wants vs what the great and powerful patron expects and spur the warlock into action or risk losing the favor (and powers) bestowed by the patron- perhaps a loved one is held hostage by the patron?  Or the warlock dies if favor is lost?  Maybe the patron sends increasingly assertive/dangerous messengers or monsters that solely engage the warlock if the character doesn't measure up?  Maybe the character seeks a way to break the pact to go back to a life of quiet research?  The patron's goals and the warlock's goals don't have to be aligned, and it's even more interesting when they're at odds than when they get along.    

It's also fair to assume that you might not need to intervene at all. If the Warlock sits back while the players lose a fight, you can expect the players to intervene, in character, with the warlock.

This type of situation depends on how it actually develops during play.

From an in-game perspective, the characters are adventuring with each other BECAUSE they generally like and trust their associates no matter what their disparate backgrounds.

If you were a character going off on dangerous adventures that could potentially be deadly at any moment in time you would NOT choose to travel with people who stand around taking notes while you risk your lives and possibly die. It's generally not how the world works. The people in your party ARE there because they will help prevent you from dying, because they will try to kill creatures who are trying to kill you or will aid you and your friends by restoring your health and containing the threat of the bad guys. If a character in the party is not contributing, there is no reason that the characters would actually consider traveling with them or splitting anything they find.

Some players come up with what they initially think of as cool character ideas. Perhaps like the detached researcher examining the motivations and effects of adventuring on the different species in the world who wants to observe but not participate in combat. However, unless this character is paying the party a lot of money to protect them (and thus being more of an NPC patron than a player character), there is no reason for the party to risk their lives escorting deadweight around.

This is the sort of conversation you may need to have with the player prior to the game to ensure that the character concept they have is compatible with being an adventurer in a party of adventurers.

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However, not being willing to kill anything doesn't immediately disqualify a character - they just need to bring enough to the group to make having the character in the party worthwhile. In one case, I know of a player with a life cleric character who would not harm another creature. They did not participate offensively in any fights but they had no qualms about blessing the party, healing the party, removing status effects ... I think the character was late tier 3 or tier 4 last I saw and was doing very well in a completely party support role and as far as I know the character had never taken an offensive action throughout their career. However, the character was worthwhile in a party because of the other abilities they were able to use that did a great job of supporting the party.

So, you need to look at each character concept on a case by case basis to make sure it works with a party - if that criterion is satisfied then it should be ok.