I found out the hard way that getting rid of fleas is a long drawn-out process. The main problem is that the fleas don't only live on the cat. They also live in the cracks along the walls and they also live on other animals nearby. So it takes a combination of techniques to really get rid of the fleas.
One of the first things you should do is to get an anti-flea floor cleaning liquid. Then clean every single place that your cat goes to in the house, which is probably just about everywhere. Then you need to bath your cat with an anti-flea shampoo.
Fleas can survive for a very long time even without food. Some can even survive for up to two years. However they must suck blood to produce eggs. So one way of breaking their breeding cycle is to deny them the blood which they need to produce the eggs. What this amounts to is that cleaning your house with the anti-flea liquid and bathing your cat with the anti-flea shampoo is something that is not just done once. You have to keep on doing it until you do not see any more fleas around. Then you just keep on doing it for, say, just as long again to be very sure. For example, if you took one month to reach a flea-free situation, then keep on using the anti-flea liquid and anti-flea shampoo for one more month.
Here's a picture to give you a better idea what you are up against: