What does "Cé leis déanamh?" mean? How is the sentence constructed? Is the meaning an idiom?

You'd need to see the context to be fully sure, but the main point is that déanamh as a noun can mean a construction or a structure or a make(as in "the make of the shoes") or a deal (in cards), so:

Cé leis déanamh = Who owns a deal = Whose deal is it/Who is to deal?

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The dialect I use is Cork Irish.
Ar sgáth a chéile a mhairid na daoine, lag agus láidir, uasal is íseal

In the context of playing cards, "Cé leis déanamh?" would mean "Who's deal is it?" or "Whose turn is it to deal?".

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I'm not a native (or entirely fluent) speaker, so be sure to wait for confirmations/corrections, especially for tattoos.