Bicycle Mechanics - Rear derailleur skipping problem

I installed a new derailleur (Deore) and I'm having a problem aligning it. When I move the mech without the cable, the gears shift perfectly, but when I attach the cable, there's a problem. When shifting towards the highest gear, the second-highest gear is omitted. When I switch to that gear, the chain just stays on the third-highest gear, and shifts to the highest gear when I go on with the shifting. When going downwards in gears, there's no such problem.
So the question is, once the limit screws are set properly, how do I adjust the cable tension?

I don't think your current problem will be resolved by adjusting cable tension, which you do with the barrel adjuster(s). There'll be one down where the cable casing contacts the derailer, perhaps one by the shifter too.
But your problem sounds like the cable isn't running freely, and maybe the cable casings not being seated properly.

+1, I don't think it's a cable tension issue since all the other gears trim OK. It could be a kink causing the cable to hang at just one point, or possibly a hanger that isn't square.

Also whenever I see an RD that shifts erratically only on the lower (larger) sprockets, I double check the "B" screw adjustment. If the RD is too high the Jockey pulley can be too close to or touching the neighboring larger sprocket making that shift problematic. Moving it down to improve clearance usually solves the problem.

The hanger is brand new and straight.

Also there's no problem with the lower gears. I had the problem you described and I already fixed it by tightening the screw.

I'll take a closer look at the cable housing in the morning, but the cable was going smoothly when I was holding my hand to it (under the bottom bracket).

Have you tried using setup guide from Shimano? http://techdocs.shimano.com/techdocs/blevel.jsp?ASSORTMENT%3C%3East_id=1408474395181679&FOLDER%3C%3Efolder_id=2534374302051135&bmUID=j7rR1Dn

You will need to work out which version of Deore you have, as there are 7 different versions of the RD available.