Not unbiblical. I wouldn't do it, but I'm not a tattoo person. I've seen very few that are breathtakingly awesome, but I wouldn't have it on me.
The idea that tattooing is varnishing the temple ---- what about digging a ditch? You get spotted up doing that, too. But, we love us some ditches don't we? If you don't have any where you are, you ain't where I'm from.
On the flipside, I do see some merit to discouraging it for a whole host of reasons. The old law of unintended consequences being a biggie. And, then there is that type and manner of tattooing that some get to be sexually provocative ---- like the bottom of the back tattoos that some gals get and then wear clothes that are worn for the purpose of showing off that tattoo. I've known several who did that and fish for male comment on it, coyly and flirtatiously. So, there is an undeniable kind of tattoo-getting and wearing that is done for purposes that are counter to the faith. Of course, the moment you say that you're a prude, etc., to some. And, that's okay, too. The problem is that you can't change out of a tattoo if you change your mind and heart. You can't really hide it if you're going to wear a swimsuit where your intent is not to be provocative. So, I can see where that would be an issue. I know someone who has had precisely that dilemma.
Plus, I don't want my daughter to get a tattoo. So, I discourage it. That's just me.
Human government bears the same relation to hell as the church bears to heaven. (David Lipscomb, On Civil Government, 72).