I have a character concept, I’m just not sure the best way to make him.

He’d be a drow from forgotten realms (or an elf from the undying court if my next game is in Eberron) who is from one of the noble houses. Unfortunately he isn’t much of a fighter or wizard, which puts his positon and life in danger

He has a streak of good in him (like drizz’t but not enough to act on). 
He makes a pack with ..... someone/something to improve his arcane abilities and secure a place in his house. However the patron takes that streak of good and influences it to get stronger until (when his mother goes birth to twins, again putting his life in danger) he decides to leave the underdark. 

what patrons and pacts would fit this story/character best? 

pact of the tome would fit a knowledge-seeking character pretty well. I´m not sure about a patron

The Celestial Patron would seem to fit that on the nose, but I have something else to suggest: What if you take Eilistraee as a reskinned Hexblade Patron? She's the only good drow god and it is her divine mission to save the souls of the drow, after all. Also, if your character is a low Intelligence (not a wizard), low Strength (not a fighter), high Charisma (someone who has gotten away with being neither of those through charm or deceit) this is kind of the chance for them to become a little bit of both, because Hexblade warlocks use their Charisma stat in melee. I think it's an interesting take.

A Celestial patron is the obvious choice, but a Great Old One whose influence twists him from evil to good sounds so fun!

Drow often turn to arcane magic for power, particularly male drow who are excluded from wielding the most potent divine magic. Many drow warlocks rechannel their early devotion to the Spider Queen to tap into the power of mighty eldritch forces. However, a significant number of surfacedwelling drow tap into their race’s long-lost past by channeling the power of the Feywild.  In the world, drow warlocks have a particularly sinister reputation, often retaining the ruthlessness of their upbringing as they advance their own agendas with little concern for others.Drow who face exile from their Underdark homes sometimes turn to warlock pacts as the best way to gain the power they need to survive on their own. Drow hexblades usually enter into pacts with archfey. The Prince of Frost is one such archfey who will make such a pact. Bestowing the "Blade of Winter's Mourning" to his bladepact warlocks. 

Drow warlocks are loners, wanderers, and mercenaries, dependent on their pact for survival in the surface world. Unfortunately for them, the typical mistrust of drow displayed by surface-dweller's world only increases when a drow is known to deal in pacts with otherworldly entities. Drow hexblades are typically pragmatists. They want power as a means to defend themselves against their murderous kin and to survive in a hostile world. The fey powers that create these pacts care little for the details of a drow’s life—a state of affairs that most exiled drow find refreshing. Drow hexblades typically worship Sehanine, though a few of the more heroic-minded ones in the Forgotten Realms embrace Eilistraee.

Any drow warlock who learned to wield mystic fey powers is a threat to the way Lolth operates. He soon discovered that the priestesses of Lolth distrusted his independence and his mysterious arcane powers. The warlock fled to seek refuge among a more civilized and possibly more tolerant world. In the forests of the surface lands he met others who loved the beauty of nature and the touch of magic as he did, friends whom he could trust with his life.  The warlock has turned his back on the hateful ways of his people. He now fights for his newfound home and the folk who have befriended him. 

- This is all inspired from 4e warlock stuff. Hopefully it gives you one possible direction for your warlock to travel down.