I did my second character as melee/unarmed.
I started with Chance's Knife directly after leaving Doc Mitchell's place and buying a shovel.
I did Sunny Smiles gun tutorial 'cause I figured shooting bottles didn't really count as using guns.
After that it was unarmed except for the knife.
Best weapons were Chance's Knife, Pushy, Cram Opener and Paladin Toaster.
I found that I really wanted ranged weapons so I allowed throwing knives and spears. Hatchets would have been OK, I just never used them.
I also used explosives but I think that was just due to lack of experience. For example, I used them in Dead Wind Cavern but I think I could now do it without blowing anything up.
I've now stopped wearing armor which makes for an extra challenge when playing melee/unarmed.
As for skills and perks, you definitely want to be very stealthy. Recon armor is useful if you aren't playing naked.