I'm working on a new character, a hexblood hexblade. I'm currently working on details like their hexblade weapon and appearance.

For the general backstory of the character, she was originally a tiefling born to Elf nobles. Her parents didn't want a "cursed" child to be their heir, so they sought help to remove the infernal mark. Their "help" turned the child into a hexblood, although the "helper" turned away for a moment and came back to a weapon laying with the child. The Elven nobles ended up taking their child back and sheltering her, trying to destroy the before mentioned weapon but eventually just locking it up. As the years went by, tension grew. She had fragmented dreams of her patron and her weapon locked away. She played along until she was informed of an arranged marriage to someone who she could tell would use her. She found her weapon and left, living in disguise.

Edit -  Updated, first person version of her backstory: I was supposed to be an elf like my parents, but I was born a tiefling. My parents didn't want a "cursed" child like me to be the heir, me being the firstborn in a noble family, but weren't going to get rid of me and try again. So they went and looked for someone who would remove the infernal mark. However, the helper turned me into this, a hexblood. During the transformation, my weapon was said to appear with me when the helper turned away. But my parents still wouldn't try again, so they raised me and locked away my weapon. Over time, they had more children, typical family expansion stuff. However, my parents were worried that I wasn't going to continue the family line even though I already found an interest, so they set up an arranged marriage. Hey, get your firstborn a partner to have kids with while raising your status and gaining benefits from allies all at once. But when I met him, I felt something wrong. A chill went up my spine whenever he spoke, I couldn't bear to look him in the eyes. I was going to immediately sneak out of home with my weapon, but I still loved my family and couldn't leave. So, I hid in the walls for a few years. I almost got caught by a niece that had heard me, but I convinced her to keep it a secret. Eventually I was able to leave home, now living a lie just to get rid of the marriage. After I left, I considered going back. Maybe my arranged partner wasn't as bad as he seemed. Even if he was how I thought, I could have told someone or went along with the marriage to keep my family's honor. But it's too late now, I guess.

I would suggest maybe a magical rapier or a magical Longsword, which kind of acts like a lightsaber, but when you press the button (or flick the switch), a flickering purple flame appears, and it is dual-wieldable and can be used with both hands. It could probably have a skull handle, with black material I’ve the hilt, adding to the gothly feel of the weapon.

I would suggest start with elf base add demonic or fiendish attributes, like horns, a tail, scales, ect. Then add hag attributes, stupped walk, moss growing on you, fingernails. I like to think of things that tell my story without telling my story. So when people first look at you they see an elf from far away. As they get closer it become noticeable that your not a regular elf. The tiefling part should be apparent next, then the hexblood. Love the back story so far. Don't get caught up in to many details, the journey forward should define rather that your past. 

my 2cp worth: For the Hexblade side of things go for an Elven Moonblade, it was the ancestral weapon of the family blessed by (insert Elven Deity) but was lost/stolen/given away several generations ago and it wants you to find it. The pact of blade feature conjures a facsimile of this weapon and this could give you a fair amount of leeway into what type of sword the moonblade is. You also then have a little roleplay with fighting off the darker sides of your nature as you struggle to prove your worth to your elven family.

This bit is maybe something for your DM to chew on: Maybe the fiend responsible for making you a Tiefling and the Hag, or other people, resonpible for then turning you into a Hexblood secretly know about the Moonblade and want it for themselves and you are a pawn they are trying to manipulate.

I agree an Elven Moonblade would be sick