Hi
To kills fleas you need to treat
both the animal and the environment at the same time
So for your cat you need to go to the vets and get a spot on treatment such as advantage, frontline etc. These are the most effective treatments for cats and as other members have pointed out flea collars are useless as are most non prescription treatments. You need to continue using this every month even in the cold weather otherwise your cat will become susceptible again.
You can also get tablets for cats and dogs called capstar - these kill any fleas on your pet pretty much straight away and can be very useful for helping get rid of a large amount of fleas before starting with your monthly treatment (I have used this when my cat came in contact with a hedgehog and I could see the fles jumping off her
>>>)
Treating the indoors is also an important aspect of flea control. Eggs will be laid in your carpet, pets bedding, soft furnishings and on your bed if your cat sleeps with you. The eggs then hatch and will jump onto your cat, reinfesting it.
For this you need to get a good insecticide - vet kem acclaim and stay kill are the best
http://www.petmeds.co.uk/c-13-cats_flea-and-tick.aspx?pagenum=2
you can get these online or from the vet. Other flea sprays from P@H for example are useless.
Vaccuum every room from top to bottom, move furniture into the middle of the room and hoover in all the edges, where the carpet meets the skirting. You then need to treat each room with an insecticide spray (don't forget curtains etc) - you don't need much just a light spray (read the instructions carefully) and remove ALL pets before using. This should be effective for between 6-12months
Make sure you treat the pets bedding with insecticide as well, a wash in the washing machine is not always effective. I always treated my carpets, curtains etc twice a year regardless of whether I saw a flea on my cat or not!
Providing you treat your cat monthly with a spot on treatment and treat your house you should find that the fleas disappear fairly quickly. Flea shampoos are not brilliant either but can be good to remove dead fleas/eggs from your pets coat and remember to brush your cat regularly to do the same.
Good Luck
x