How do I go about cleaning up Riften or is it actually not possible? It seems all hope is lost for the citizens of Riften, but they still have me!
There are a number of things behind the state of Riften: some are solvable, others are Riften-by-design. Riften is full of grays, and you might have to modify your definition of "success" while trying to improve the state of things there.
Dealing with the Skooma trade
Riften's full of junkies. You can help them get above the influence by disrupting the Skooma trade in the Rift (adapted from my answer about getting a house there):
Improving the quality of life of the citizenry
With all the worrying and fretting about their conditions in Riften, a lot of citizens just need a helping hand. Ask around and see if you can help people out: by completing miscellaneous objectives, people tend to act a little more up-beat.
The Honorhall Orphanage
The head of the orphanage, Grelod the Kind, does not live up to her name. She keeps the orphans in a state of fear and is all around not a good person. You can think of the little children and confront Grelod:
The Thieves' Guild
There's a certain group Riftenites who are also down on their luck: the Thieves' Guild. Once an establishment to be feared across Skyrim and Tamriel, it's now been reduced to a gang of petty thugs and criminals.
You can get them back to their former glory by helping them take care of business and eradicate their competition in the Rift. You might run into Brynjolf during your time in Riften, who will begin the quest chain for them.
You can also help their main benefactor take over the Rift during the Civil War:
Certain parts of Riften are clean-able, but the Thieves Guild and Black Briars are here to stay.
See @MarkTrapp's answer for more clarification of what you can do to help the Riftenites.