Yeah, it sounds like you are using metonymy, substituting the word "ink" for "tattoo". You have to be very careful with this when translating into another language, as there is no guarantee that the same word relationship holds in the second language. In other words: you should say precisely what you mean before translating.
As it turns out, there are some tattoo shops in Japan which contain "ink" in their name -- possibly because of western influence. However, judging from a google image search, I don't think that the word "ink" is commonly used to mean "tattoo" in Japanese.
If your objective is to obtain a set of Japanese writing characters to convey the idea "addicted to tattoos" or to translate the name of a tattoo shop called Ink Addiction into Japanese, I would recommend the following:
刺青中毒 (irezumi chudoku)
This expression uses all kanji characters -- two for each word, and has a nice look and balance to it, I think.
There are lots of options, though. There are usually a number of ways to translate an expression into another language.