Don't just screw around with the barrel adjuster.

Here's what you do:

first, flip the bike over.

(1) shift to the smallest cog.
(2) release the cable.
(3) turn the barrel adjuster so it's all the way in (e.g. minimum tension), then one click out.
(4) using pliers, pull the cable tight (e.g. pre-tension) and then tighten the cable fixing bolt (5-7nm)
(5) shift one cog up. (the chain won't go, don't worry).
(6) add tension to the barrel adjuster until it shifts up to the next cog, then another ~4 clicks.
(7) look closely at where the chain is sitting between the cogs (the bike is flipped over, so you're looking at the bottom of the cassette). it should be sitting so there's equal space between the cog below and the cog above. use the barrel adjuster to add or release tension until it's centered.
(8) shift through the cassette. shifts should be instant, and there should be no rhythmic clicking. check that the chain is centered on each cog.

easy-peasy.