You get the quest "Taking Care of Business" from Byrnjolf after framing Brand-Shei for stealing Madesi's Silver Ring as part of the quest "A Chance Arrangement" in the Riften Market.

Open your quest journal:
Taking Care of Business
Brynjolf has promised me more work if I'm able to locate him at a tavern called the Ragged Flagon. Judging from the dangerous nature of its location, it seems as though he's having me perform some type of initiation, but I can't be certain.

Locate Brynjolf at the Ragged Flagon

Go down the wooden stairs south of the market. Then find the gate to the north. Open the gate, then go through the door. Descend down the tunnel. Work your way through the corridors. You can either kill, or sneak by, the Drahff patrols there. Find the Ragged Flagon tavern towards the east of the Ratways.

Talk to Brynjolf. Tell him getting there was easy. He will ask you to handle a few deadbeats for him. They owe the Thieves Guild some coin and decided not to pay. He wants you to "explain" to them the error in their ways. Tell him it sounds good and ask who they are. He will tell you they are Keerava, Bersi Honey-Hand, and Haelga. If you do this right, he promises you a permanent place in their organization. Ask him how he wants you to handle this. He will say that the debt is secondary, more important is the message that the Thieves Guild will not be ignored. But he does not want any of them killed since it's bad for business. Ask him if you will get a cut, and he says that you will since they take care of their own.

Open your quest log:
"After successfullly locating the Ragged Flagon, Byrnjolf has given me a new assignment. Three business owners in Riften are proving to be a thorn in his side and I'm supposed to teach them the error of their ways. I should speak to Byrnjolf about these townspeople in the event that they prove to be as stubborn as I'm expecting them to be."

Collect Haelga's Debt
Collect Bersi Honey-Hand's Debt
Collect Keerava's Debt

Ask him for tips. For Bersi, the key is a dwarven urn in his shop. If you smash it to bits he will change his attitude. For Haelga, use the statue of the divine as leverage. Keerava has a soft spot for family, so talk to Talen-Jei at the Bee and Barb.  

You will now see these optional quest objectives:
Use Talen-Jei to get to Keerava
Steal Haelga's Statue of Dibella
Smash Bersi's prized Dwarven Urn

Exit the Ratway back up to Riften. Enter the Bee and the Barb just north of the Market. Locate Talen-Jei. Tell him to talk some sense into Keerava. Ask him to help you convince her to pay. He wants to marry her and cares about her deeply so he does not want to see her hurt. He will tell you she has family in Morrowind and that if you mention that family she will pay you.

Go talk to keerava behind the bar. Ask her if she is ready to pay Brynjolf what she owes. She will say she never will. Tell her you will pay a visit to the farm in Morrowind. She will beg you not to hurt her family. Tell her to pay up and you'll forget about it. She will pay you  the debt.

Exit the tavern. Enter the Pawned Prawn. Tell Bersi you have a message from Brynjolf. Tell him if he doesn't pay, bad things happen. Tell him you've had enough of this. Find the golden dwarven urn on the wall to the east. Start smashing it with your weapon. Keep hitting the Urn. He will beg you not to destroy it. Ignore him, and keep going at it. After it breaks, ask him if he wants you to break anything else. He will say he will pay on time from now on, and give you the 100 gold he owes. 

Next, go to Haelga's Bunkhouse to the north. Tell her you have a message from Brynjolf. If she is the last person you are visiting, she might have heard about how you terrorized the others, and pay you the debt right away  out of fear. Otherwise, tell her it's not about the money anymore. She will ask you to be reasonable, that she does not have the money and will pay next month.

Sneak and steal the statue of  Dibella near the wall. Then talk to Haelga again and threaten to throw it down a well. She will give you the gold. Then go back to the Ratway and return to Brynjolf at the Ragged Flagon. He will be happy you kept it clean without needing to kill anyone. Give him the gold you collected. He will reward you with some potions and poisons. Ask him what's next. He will say they need more people like you. Tell him that if there's more gold from where that came from, you are in.

"I've dealt with the three troublesome business owners in Riften as Brynolf had specified. Perhaps now that I've proven my worth to Brynjolf, I can discover more about this mysterious organization that he represents"

He will ask you to follow him so that he can show you what the Thieves Guild is all about.

This completes the quest "Taking Care of Business" and starts the next Thieves Guild Quest, "Loud and Clear".