Yes DKs can heal - Quick Siphon, Healing Ward, Healing Springs and Energy Orbs are all you need

Trials and Dungeon set
Spell Power Cure / Twilight Remedy / 1 Piece Rkugamz

Quick Siphon - Everyone heals themselves just for attacking
Healing Ward - Your Oh-s... heal
Healing Springs - For the heavy incoming damage
Energy Orbs - Your heal over time (Orbs, Orb synergy, Twilight synergy
Igneous Shield - Major Mending, cast it every 6 seconds
- Pour CP into Baston to make this a big shield too

Cauterize replaces Inner Light on the back bar for and additional healing there and it gives you Major Prophecy as well
Wall of Elements and Eruption are good DoTs which you run solo, and dont need to be refreshed when incoming damage is high.