If you're cat has got fleas, and you think they may have also infested in your home you need to get this sorted as soon as possible.
Firstly, I would recommend that you get to grips with what your up against. Fleas begin as eggs, which can be laid on furniture and in the pet's bed, for example. The female adult flea can lay up to 50 eggs a day, which can fall off onto carpets, floors and clothes.They then hatch into larvae, which are actively small maggots. The larvae feed off adult flea's faeces and other matter found in carpet fibres. The larvae spin themselves a cocoon which forms a pupae and eventually hatches out again into an adult flea some 8-14 days later.
Fleas favour warm environments and are attracted to the heat of a body and moist climates. Fleas feed off their host, sucking blood and laying eggs (females).
The first thing you need to do is take your cat to the vets. They will adminster treatment and although this is not always effective in eliminating all fleas, it will reduce numbers for you. Don't get rid of your pet or the fleas will simply start to feed on you.
You need to treat the area where the cat lives - if he runs about the whole house I would recommend a deep clean. Wash all linen and cushions, sofa covers etc on a hot wash (over 60C) to kill any eggs that are on them. You might want to hire a steam cleaner and wash all your carpets thoroughly, as vacumming has little effect on fleas. You could also spray insecticide on the carpets to kill off any adults (it won't kill eggs so keep up the spraying until they've all hatched).
Continue to be vigilant with your cleaning routine and perhaps keep your cat in one room only, so that the fleas do not keep spreading everywhere.
Follow Blurto's instruction to a tee and start putting garlic (small unnoticeable amounts at first in your cats food, increasing as the days go by until you have about a clove in each meal) this will help your cat with many infestations from fleas to ticks to worms, it works, but be gentle on your cat as his constitution is much smaller and more tender than yours.
Putting moth balls or flakes into vacuum cleaner bag, as you vacuum the fleas, and eggs that are picked up will suffocate in the bag. I keep moth balls in my vacuum cleaner year round and it works with ants and any other bugs.
moth balls should NEVER be used around cats as they are TOXIC to them. please use a flea collar in the bag or borax powder or diatomaceous earth to kill fleas instead--very effective and much safer.