So yeah, I thought we had enough 'how did you kill Benny?' threads and stories about how you killed a giant abomination with horns that kills lvl 30 Couriers with one hit are generally more fun then stories about how you killed Chandler. To me, the weapon I used to kill the Legendary Deathclaw (LD) has from now on a special place in my heart, while the one I used to kill Benny doesn't.

Anyway: I was just sneaking around in the Dead Wind Cavern when I was like 'omg am I really gonna kill the LD with the AER14 omgomgomg', so then I used V.A.T.S. to get a sneak attack critical to the head, and then I shot him some more. I have a freaking high crit rate (9 Luck + Built to Destroy + Finesse + Light Touch + 1st Recon beret) so eventually the LD became dust c: . That day I hotkeyed the AER14 to 1 and got the Laser Commander perk.

Anti-material Rifle and AP rounds. One shot.

I was unarmed at lvl 12 with power fist so I just lobbed my holy hand grenades at him.
Haven't been in there with another character yet.
Probably gonna take him out by placing some mines and stuff to cripple the legs then kill him with RCW+.

Ranger Sequoia, HP rounds, and a sneak attack to the head.

Sneak at 100, Luck at 10. Every critical enhancing perk, every perk associated with sneaking.

Sniper rifle with all mods, using jacketed soft-point rounds. And nothing but sneak attacks, no psycho or any other chems. One shot to the head.

Next time I'm gonna use the Brush Gun modified. I find it to be better than the Anti-Materiel. And the ammo is cheaper and easier to come by.

Assault Carbine and ap rounds.

Some mines wouldn't hurt.

I shot it with a brush gun. That gun is so OP.

A 50. caliber bullet to the face. Fast and clean. ( well asside from all the those little bits and pieces of brain on the floor)

The best way to kill a Legendary Deathclaw is with a Legendary Rolling Pin.

2 or 3 shots with Alien blaster and boom.

how about the alien blaster

I confused him with a girly shriek of pants-wetting terror and then shot him in the head with my Trail Carbine. Then I tired him out by making him chase me. Then I shot him in the head with the Trail Carbine again. Then I lulled him into a false sense of security by almost dying from being gutted. Then I shot him in the head with the Trail Carbine. Then I heroically stopped time and injected myself with multiple Stims; re-entering the normal flow of the universe, I shot him -- in the head -- with my Trail Carbine. He died.
Easy-peasy.

Crapload of mines, sniper rifle, and Boone.

I've only ever gone there while I was level 10-12 in which I sort of cheated. I always use the Incinerator from the Prison Leader in the Bison Steve.

Step 1 - Go to the cave and hide behind the rock next to the entrance killing any Deathclaw outside.
Step 2 - Go in cave an lure out the tougher Deathclaw (I got lucky once and they all came out)
Step 3 - Repeat step one
Step 4 - Kill all the Deathclaw that followed you
Repeat Steps 2-4 if more Deathclaw remain.

I have to say the incinerator also works nicely on Deathclaw Promontory which at level 10-12 is a nice way to get Enclave and Brotherhood armor...too bad you can't wear it yet <_< (Well the way I play you can't)