I've written a program to generate a fractal, but my computer would take a week to compute it. Is there any online or otherwise, solution where I can run my code and have it compute quickly?
No, not to be rude and don't take this wrong. Nobody is going to let people use there full computer's resources over the web to let you run code, it will always be slower online. For free. However VPS and Cloud-Computing options are available. They will be expensive for a really fast computer.