How to complete my death claw objective?

Deathclaw attacks will not start until your population raches 60. It is a random event , made more likely by having an active radio station and /or opening the vault door. They are extremely dangerous to rookie players. The carnage will shock you.

Once you have learned how to stop them, and depending on your vault this is usually a multi-room undertaking, with rare quality weapons and outfits being nearly essential, you can weaken the last room a bit so that the last, beatup deathclaw moves on to another room which should be full of lvl 50 unarmed dwellers in good outfits. If you get it right - and with practice you will- you can safely kill one without losing dwellers. Rinse and repeat. It takes a bit to get these tasks done , but they are well worth it paying one sort of box or another.