According to some Orthodox Jews, the presence of a tattoo actually makes the body unfit for Jewish burial and even unfit for participation in a synagogue service! This is an extreme view, however, not shared by the majority of Jews. Nevertheless, the consensus of Jewish law and tradition is firm on this subject: tattoos are prohibited.
If you are a Christian or Messianic Jew, then you are not under Rabbinical authority (B"H), but nonetheless you might want to keep in mind that you are to be a walking testimony of the LORD and His impact on your life. How a tattoo fits into that testimony is, of course, between you and the Master. However, if you are hoping to be a witness to the Jewish people or to walk out your life as a spiritual Jew, you need to be aware that your tattoos would be generally offensive to Jewish religious sentiment.
Of all the tattoos that would be considered offensive, however, a tattoo with the Hebrew Name of God (or any derivative thereof) is surely the worst, since Jewish halakhah and tradition have very strict rules regarding the writing of the Name (to keep it from physical desecration, to keep it from being used in vain, and so on).
For those of you who might be considering getting a tattoo, I would simply suggest that you pray about your decision - and keep an open mind to hear from the Spirit of God. But if you are a tattoo artist or someone hoping to get the Hebrew lettering for a tattoo you want to put on your body, please find someone else to help you, since my conviction in this matter is to abstain from this practice.
A final note:
Please let me add a closing note of comfort to those of you who have tattoos. While it's true that Jewish tradition prohibits getting them, there are many people that love and honor the LORD God of Israel who have tattoos. Please understand that this article is not meant to bring any sense of shame or condemnation upon you. My intent is simply to inform those of you who might be considering getting a tattoo that doing so is contrary to the Jewish way of thinking about the human body as b'tzelem elohim. Nonetheless, just as the Mashiach Yeshua was "marred" and disfigured for our transgressions (Isa 52:14), so He is forever able (and willing) to completely make those who trust in Him to be "whiter than snow" (Isa 1:18; Ps 51:7). God be with you, chaverim.