Take your cat to the vet and get Bravecto. It's prescription-only, which means that your vet needs to examine your cat at least once per year to prescribe it. It only needs to be applied quarterly.
You will also need Drontal wormer every six months.
Once your cat is treated with Bravecto, let him roam freely throughout the house. A treated cat is your number one weapon against the fleas in your house because they will hop into him, bite once, and die. You can hurry that process along a bit with a flea fogger in each room, but shut both the cat and the humans out of the room whilst the fogger is running and for several hours afterwards.
Fleas have a four-stage lifecycle, like butterflies have: adult lay eggs, which hatch into larvae, the larvae then pupate, and finally the pupae emerge as adults. The foggers and room sprays kill eggs and larvae and also contain hormones that prevent larvae from turning into pupae. These products can't harm pupae though: pupae are pretty much bomb-proof and it's only when they emerge into adults that they can be killed. It's emerging adults that die by hopping onto your cat and taking a bite.
Don't waste your time and money on flea "treatments" from shops, even Pets At Home. All the not-prescription stuff has stopped working because the fleas have developed resistance to it.