With all due sympathy, it sounds like your technique could do with a little refining.

One the couple of occasions that I've attempted feral-moggicide (is that the right term?) bare handed, I've found the following helps.
(1) Grasp the nettle FIRMLY. You WILL get claws - and possibly teeth - through your tender pink hide, so the tactic is to minimise repeated raking. Do not muck about trying to get the perfect grip, but wrap two big hands around as much of the cat as possible (one around the neck is good) and shut down as hard as you can. The aim is to minimise the cat's ability to move.
(2) Using knees or any other suitable part of your anatomy, apply as much weight as possible to any part of the cat not already being held in your best immitation of a vice-like grip.
(3) Cats are tough. Don't bother trying to break its neck with some exotic brand of Kung Fu.... 50+ kilos of pressure via one of the afore-mentioned knees on the rib-cage will induce a very satisfactory state of suffocation in a few minutes, and at far less risk than trying anything more complicated. With foxes, I take them by the back legs and beat their head against the nearest solid object. Much quicker, but foxes lack a cat's ability to reinsert claws faster than you can pry them out. .

Pry the claws out of your extremities after the cat is dead. Very dead.

Gloves are better.

Shotguns are best of all.