Melee/unarmed is very doable, so long as you get the right gear. Not much can stand up to the ballistic fist. However, fighting deathclaws is incredibly dicey, it's ok one-on-one but if they outnumber you then it can turn very ugly very quickly. Super Slam and Ranger Takedown can help level the playing field though, and if all else fails there's always the unarmed character's doomsday device: boxing gloves. Honestly, I'm not kidding, deathclaws don't have nearly as much fatigue as they do HP.
But yeah, I completed a CL playthrough using strictly only unarmed/melee, and deathclaws aside it was pretty straightforward. Pick your battles and fight smart early on, and by level 15+ you'll be practically unstoppable. Pushy and the Paladin Toaster were my weapons of choice for the end-game, in the early game a set of spiked knuckles did me just fine.
Oh! and invest in either survival or medicine. You will be taking hits.