I created a melee-focused character, when played New Vegas for the first time ( soon after release ). It was absolutely fine ( on hard difficulty ), but I don't recall any details. Simply a different combat style. And you don't need to think about ammo. ;)

It is fully viable. The trick to fighting Deathclaws is to max out your DR. Loading up on various combat drugs makes you nigh immortal. Logan's Run can be useful in that regard. A Ripper or Chainsaw or Thermic Lance makes quick work of them.

What drugs are you talking about? And what if you don't take Logan's Loophole? What else do you need to survive against Deathclaws, especially those in the Courier's Mile?

There are many drugs with worthwhile combat effects, but for damage resistance you have Med-X, Slasher, and Battle Brew. With the highest damage resistance, your damage threshold is no longer significant, so you can use lighter armors and move faster.

If you don't take the Loophole, you're bound to become addicted, which is a huge hassle. Fixer can help temporarily, but you can become addicted to it as well. Doctors can treat addiction but sometimes there are none within easy access. I was able to get all the perks I cared about despite the lower maximum level imposed by Loophole.

Against the most dangerous Deathclaws, a dose of Turbo is very useful. Bursting them down is best, so the usual crit builds are ideal. Taking Survival and Science skills is a good idea. You can craft drugs, healing foods, and the Snakebite Tourniquet which is great for cazadors and other venomous creatures.

Staying friendly with the Khans gives you another good source for drugs.

What do you mean by 'bursting them down'?

Damaging them faster than they damage you.

And how is that done?

As I said, weapons like the Ripper, Chainsaw, and Thermic Lance are great. Plus buffing yourself with Buffout or Psycho, maybe even some Jet. If you're not using an "automatic" weapon like the above, then Rushing Water adds a nice attack speed bonus.

That's very helpful advice.

So basically, what you're saying is that I should make Unarmed or Melee builds, not those that use both ranged and melee. Rather, I should just pick one and stick with it?

Is there an advantage to using both unarmed and melee weapons in one build or should I just focus on one, even if they use many of the same perks?

If you want to combine ranged and melee that's fine, but it's not required by any means. I did unarmed and melee together in one build since I was able to fit everything I wanted in it, and I didn't expect them to offer drastically different experiences.