Fallout 4: Deal with a Deathclaw

Although it's possible to defeat the deathclaw through conventional melee- or weapons combat, that requires at least a moderate-level character with reasonably powerful weapons and upgrades.

I've gone through this quest 4-5 times and always use this strategy. This won't let you take out the deathclaw as a level 5 character, but it lets you win the encounter with a lower level character than using a conventional head-to-head combat approach.

1. Bring mines and grenades. Lots of mines and grenades.
2. From the top of the stair, sneak into the room and lay the mines all in one spot. Make sure they're far enough into the room that the blast won't reach into the stairwell. (The blast won't go around a corner to get into the stairwell.)
3. Provoke the deathclaw then RUN back to the stairwell. The deathclaw can't reach you there (unless she glitches through the wall).
4. After taking damage, the deathclaw will start wandering back to her nest. Peak out, throw a grenade, and hide again. Repeat step 4 as needed.

If you run out of grenades you can repeatedly peak around the corner for 1-2 shots with whatever weapons you happen to have on you, aiming for her head to maximize damage. But this takes a long time. Your best bet for a quick resolution is explosives and cowardice.

Just like almost everything else in the game, you should shoot the deathclaw in the head with crits until it dies.

This requires that your character has the appropriate perks, mostly the luck ones.

With the Deliverer, most of the luck perks, and gear that reduces AP costs, I can fire 20 rounds in a single bar of AP, and every third shot or so is a crit. If that doesn't kill my target, then a single crit with Kellog's Pistol completely refills my AP bar and we go again.

Add some Psycho and/or Jet to that, and even an on-level deathclaw will die before it closes the distance.

Like I said, that works on pretty much everything. In my last few playthroughs, I killed the courser in the Greenetech building on my first bar of AP before he got a shot off.

If you didn't set your character up in an effective manner, then hiding and trying to blow it up will probably work, as described in Feral Chimp's answer.

Personally, I handle Deathclaws (and all other big mobs) the same way:

1. Fully upgrade the Combat Shotgun for maximum damage
2. Pop PsychoJet & Med-X
3. Get just out of melee range and unload the magazineinto their head
4. Repeat step 3 until dead

With this strategy at level 35+, I haven't had much last longer than the Jet, and if they do, one or two reloads afterwards and they're a loot pinata. I don't have a Legendary shotgun, and I think I've got only 1 rank in Rifleman, if that.

you can always find a place death claw cant reach you. a small door will do in this place. hide there and peak and hit. scale your damage up using phyco and other drugs if you want to speed up the process. mines are great in such situation too