Scripture's condemnation of tattoos (Le 19:28) is in the middle of a passage condemning obvious sins as well as outwardly harmless things for reasons that may be entirely lost to us historically:
You shall not:
26: eat anything with the blood
26: practice divination or soothsaying.
27: round off the side-growth of your heads
27: harm the edges of your beard.
28: make any cuts in your body for the dead
28: make any tattoo marks on yourselves:
My first thought is that, like many Levitical laws, these were meant to separate Israel from pagan nations. Holiness in Judaism was and is strongly based upon separation - the idea of something being holy is coequal with something being dedicated for the worship of the Lord. Paul twice uses the phrase "set apart" to describe himself and his calling to evangelize the Gospel (Ro 1:1, Ga 1:15).
In that light, the admonitions in blue above make more sense if we can demonstrate that there were pagan practices associated with marking the body or cutting the hair. To avoid adopting these practices meant visibly calling oneself for the Lord, or at least refusing to affiliate with a pagan cult that had adopted the practice. Obviously divination is harmful as is self-mutilation, so these proscriptions make sense and should remain for us as Catholics. But given that the association with tattoos and shorn beards has long since passed, I think that getting tattoos is not intrinsically evil.
That said, I'd argue against getting one on several grounds:
1) If it is viewed as a marker of gang affiliation or advocacy towards the occult or violence. I don't think you'd be asking if you were going to have a crown over an upside-down trident tattooed on your arm, but it should be said.
2) If it conveys a sexual message. Showing off a so-called "tramp stamp" is socially believed to convey that the women bearing it is "easy". This goes along with any concerns of modest dress or behavior.
3) If it is dangerous to do so - if you're getting one in a place where sanitation isn't up to snuff you're taking a risk. Before I got mine I inspected the autoclave, the single-use needle, etc.
4) If it will cause some kind of detriment to your future. Getting a tattoo on the side of your neck severely limits your employment options. If you wanna know, mine's on my bicep where a dress shirt covers it. I actually forget it's there unless I brush my teeth right after a shower and happen to look in the mirror.
Here's the biggie for me:
5) If you're an eligible blood donor. Blood supplies are constantly low and eligible rolls drop daily because of infection and declining health of the donors. I'd almost say that, if you're healthy and can do so, you have a duty to donate blood at least a couple of times a year. I was a frequent donor but following a diagnosis of celiac disease I've been told not to, and it honestly pains me.
I hope that answers your question.