He needs to see a vet. I can't really tell from the picture what is wrong but a vet should be able to tell with an exam. A vet would also be able to treat it more effectively and with less chance of complications than any home treatment. If you are right about it being an ingrown claw you can ask your vet what signs would be to allow for home treatment and how to treat it.
I know its probably not the answer you were looking for. But, cats hide pain and injury. If your cat is limping it has likely gone beyond the beginning of a problem and is in full blown Problem mode. He likely needs antibiotics to prevent or treat infection and prevent it from causing additional problems. Or it could be something other than an ingrown toenail and require different treatment all together.
Hopefully your vet can give you signs and steps to take to monitor if he is prone to ingrown claws and help handle them before they get bad. So that jext time you can handle it at home before your cat is limping. Assuming it is that.