One of my players backstories has them possessed by a demon, for my adventure I was thinking of having the players try to fight off war bands of gnolls which seem to be targeting weaker villages. The players find that the gnolls have cultist among them and demons. The players track the gnoll bands back to cave that leads to a abyssal rift. The gnolls are following the orders of a warlock of Yeenghou, who keeps summoning demons through the rift. When the players arrive the pc possessed by the demon will feel the demon slowly take over as she draws closer to the rift. Is this good enough plot? 

It has some merit.  

Questions you want to consider:  How does the player maintain a sense of agency?  "You feel compelled to do X" can really wear down a player and make them feel helpless.  Good in small doses, lethal in large.  Like Whiskey.

I would consider a "win state" for the party and leave yourself as open as possible to the dozens of ways they'll try to succeed.  Do they need to close the portal?  Go through and kill the demon?  Also, be careful of "well they went through so OF COURSE she's totally possessed now".  If you're going to go there you'll want to really warn them somehow of that possibility.  Players can often miss hits and the last thing you want is a player unable to play for long stretches at your table.

I ran a character once where his backstory was that he botched making a pact with a demon to become a warlock and ended up binding the demons soul into his body , making him a wild magic sorcerer. The demon could only be freed if the character died. The character had this storyline of trying to figure out how to reverse the binding without dying and the demon was forever trying to kill the character. This manifested not by the DM telling me I have to do stuff, but rather by having the demon influence what was happening around me. He did this through the occasional unexpected advantage or disadvantage, spell effects, the wild magic effects, etc. This worked well, without me losing any agency. You could use a similar mechanism, but have it escalate in magnitude and impact as things progress. 

Was..I think we are talking about the same person. Did they play a Tiefling? 

Does the rest of the party know?

If they’re all on board and want to help, there could be a quest for something, an item or spell to maybe just temporarily block the control of the demon. They activate the relic and now they have 1 day to kill the demon and end the curse or it takes total control. 

Maybe the PC gets disadvantage against the demon's allies, and as the PC gets closer to the rift it can't attack them at all. The PC might also get noticably more violent, and might destroy things for no reason.