Both melee and unarmed are viable for all but a few situations.
You will want a weapon for very long range killing. Grenade rifle, Gauss Rifle, Sniper rifle of choice. You don't have to be perfect with it, you just need something to reach out and touch things you just CAN'T close on. That, and it helps thin out the herd.
Sneak is your friend. Take silent running, get it HIGH. Low to medium sneak is discouraging. Max sneak is insanely fun (not as fun as FO3 with the chinese stealth suit, but fun).
You don't want to melee deathclaws. You THINK you want to, but you don't. You CAN kill them without taking damage. It can just be an exercise in frustration if you time things wrong and oops, time to reaload. There are plenty of other ways to make deathclaws die. On the other hand, I have found no better way to kill a Cazador than a super heated saturnite fist to the schnoz.
Corners are your friend. Make them come screaming around the corner ... right into your fist/weapon.
Melee and unarmed weapons deal VATS damage that puts almost everything else to shame, due to the fact that they ALL do double damage in VATS. Use it, abuse it, especially with unarmed.
For unarmed, get 75 skill so you can use the 'cross' attack ASAP. It's downright brutal. Melee is a bit more complex, but there are skill levels at which very fun special attacks get unlocked (varies by weapon for melee).
tl:dr Sneak, silent running, corners, have a very long ranged weapon.