I would say that tattoos, in general, are morally neutral.
Regarding the idea of our bodies not belonging entirely to us but to God: I am not certain this is exactly accurate. Our bodies are a gift from God, therefore creating an obligation in us not to misuse them. But they are ours more than anything else we have, unless you would contradict the teaching of the Church and say we have no right to private property? And if we can have dominion over things not a part of us, how much more do we own our bodies?
Besides which, in Aristotelian philosophy, even if there were no God, it is wrong to misuse anything. Owning it does not change that fact–ownership simply imparts, among other things, the right to use it at all.
Anyway, as to what tattoos are wrong and therefore constitute an abuse of the body: I would say that tattoos associated with gangs are wrong, especially those of gangs like MS13, the Aryan Brotherhood (now the Brand), or the Russian mafia. Even more so, the elaborate systems of tattoos used by Asian organized crime, like the Chinese Triads, the Japanese yakuza (full-body tattoos of mytholigical figures, often very beautiful) and the Korean kkangpae (designs incorporating roses).
Apart from crimes symbolized by the tattoos themselves, the tattoos mark one out as a criminal, separate from society, and implicitly swear allegiance to the way of life of the gangster. The yakuza code of jingi (benevolent justice), though not without a certain Stoic nobility, is entirely incompatible with Christianity.