I guess both computer skills and computing skills are correct, depend on your purpose. Computer skills means the skills about computer aplication, and computing skills means the skills to compute, and have no relation with computer. Computer itself was mean machine to compute.
I hope this could help.

Computing skill = Skill in processing information, i.e. in using (computer) applications. Software emphasis.

Computer skill = Skill in computer technology, i.e. in using, maintaining, installing and perhaps even in building computers. Hardware emphasis.

The skills associated with the use of computers are most commonly called "computer skills." Our new secretary is computer-savvy. She knows how back up files, trouble-shoot application errors, create fancy e-mail, customize installation of software, upgrade hardware, etc. She has a lot of "computer skills."

On the other hand, "Computing" tends to be used by techies, and typically refers to what computers actually do. One speaks of the "evolving computing needs of this organization," or the "combined computing power of a network," or the "computing challenge that our expansin will create," or "new computing technologies."

The term is less used in the context of skills, although I can imagine someone speaking of skills in dealing with "computing problems or issues." It would not be wrong to say, "computing skills;" it would introduce an emphasis that in most situations would be somewhat odd, but that may be appropriate in some contexts, as when we attribute such skills to a developer, for instance.

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