I forgot to mention that there are many groomers that come to the house which will eliminate the stress for your cat.
You should be able to cut a thick claw with a dog nail clipper, but if the claw has already grown into the pad your cat will most likely require antibiotics, so a visit to the vet is necessary.
Vets don't charge that much for nail clips, and you can get your cat's anti-biotics there.
Many groomers will clip nails for a nominal fee. Please take your cat for a nail clip well before it gets to this point. Nail clipping once per month should take care of the problem.