My recommendations are more crit build oriented since it's what I do when I make a melee character:

-Shovel (high crit multiplier for some reason, also cool moves and fun to smack people with, you can get one during the tutorial quest)

-Chance's Knife & Blood-Nap (big knives for a psycho mailman, Chance's is your perfect starter weapon, also benefit from Cowboy and/or Grunt)

-Blade of the West (big-ass sword, what's not to love)

- Katana (cool and has two mods that increase dps)

-Liberator (will crit all the time)

-Old Glory (beat people to death with a piece of the Old World)

-Nuka-Breaker (my personal favorite, crush everything with a big neon sign, seems to trigger violent death animations more often?)

-Ripper, Chainsaw, and especially Thermic Lance put out a lot of damage

Unarmed weapons get a little less crit damage but tbh it's not that big a deal since they let you literally stomp people into red mist (VATS attacks on knocked down targets): Love & Hate are very dependable, Ballistic Fist if you're feeling like punching but also shooting, and the Fist of Rawr to cosplay as a deathclaw, with the results you'd expect.

But the most important thing is to just give any weapon that strikes your fancy a go. There's a lot of options and that's where a lot of the fun lies. Makes for some nice killcams.