How do I buy the house in Riften?

Who do I talk to to start the quest to buy the house in Riften? I have talked the Steward, the Jarl, and did a quest for the court mage, and still no quest started. Do I have to kill the dragon behind Riften, or is there some other way to do this?

This is done in three parts:

Part 1: Get into the good graces of the Jarl

How you do this is dependent on who controls the city during the Civil war:

1. If you don't let the Imperials take control of the city, the Jarl will be Laila Law-Giver. You can win her favor by completing all parts of the miscellaneous objectives that starts with Wujeeta at the Fishery.

2. If the Imperials have taken control, you can win favor with the newly installed Jarl by speaking to them.

Part 2: Become a champion of the people of Riften

Once the Jarl likes you, you'll be able to secure her trust by helping the people of Riften. Completing five miscellaneous objectives or favors in Riften will complete this step.

Part 3: Handle Riften's drug problem

There are three miscellaneous objectives you need to complete: "Helping Hand", "The Raid", and "Supply and Demand". You might've already completed these as part of the first two steps.

Once complete, you can purchase your new home for a cool 8,000 gold.

For a complete walkthrough, this is exactly what I did:

1. Spoke to Wujeeta and completed "Helping Hand" by giving her a healing potion.
2. Intimidated Wujeeta into telling me where she got her Skooma
3. Spoke to Jarl Laila in Mistveil Keep and told her who the dealer is, and accepted her request to shut him down
4. Went into the Riften Warehouse, confronted the dealer, and retrieved the information about where the Skooma was coming from
5. Told the Jarl about the source, who then told me to deal with it (noting that I'd be well on my way to a title in Riften for doing so)
6. Went north to Craigslane Cavern and disrupted the Skooma operation by killing everyone inside and outside
7. Reported back to the Jarl about the disruption. At this point, the Jarl didn't have anything new to say.
8. Spoke to Wylandriah, the Jarl's mage, and completed her retrieval quests by going to Windhelm, Winterhold, and Ivarstead.
9. Spoke to Sibbi Black-Briar in the Riften jail, and subsequently completed "The Lovers Requital"
10. Completed a dragon bounty given by the Jarl's steward.
11. Spoke to Romlyn Dreth and subsequently completed "Under the Table."

At this point, I received a new objective to speak to the Jarl. Upon speaking to her, she said she'd make me Thane once I purchased a house in Riften, which I was then allowed to do by speaking to the Steward.

You need to do as many quests as possible in the Riften area, particularly the quest that Wujeeta gives you who is at the Riften Fishery. You need to follow all of that quest line through to the end.

You also need to do quests for the people of Riften and for the Jarl. After you have done everything you should be able to buy the house.