While I kinda agree with Inky.. I also kind don't

Sorry!
Although in theory there should be no danger in getting tattooed while breasfeeding.. there are dangers so please be careful.
Make sure you know enough about the tattooing process to know that the studio you're going to is safe. There is NO standard regulation in the tattoo industry and ANYONE can get a kit off ebay and open a shop. Yes, you have to register with your local council, and yes the Environmental Helath dude may show up. Will he be able to see on one visit if the autoclave is being used correctly every time? No. Does he know enough about the tattoo industy to be able to spot the signs of bad practice in a shop that looks spotless? Probably not.
Make sure the artist is more than competent. Ask for photos of their work, not just artwork on the walls. Go by personal recommendation.. ask people you now that have lots of tattoo coverage where they go. ANYONE can say they are a tattooist.. there are no exams to pass, courses to go on.. ANYONE can do it. Not a good thing.
Tattoo inks are generally safe.. although it is more common than you might think for people to have a bad reaction to certain inks.. particularly reds. There's also not specific regulation regarding the formulation of tattoo inks (at least there def wasn't a year ago.. can't see that much has changed since then that we wouldn't know about.) Yes the ingredients have to be passed as safe to humans.. but not specifically for insertion under the skin.
Now to be fair these points apply to EVRYONE getting a tattoo.. not just those with new babies. But please bear in mind that firstly your body is still healing after the trauma of chidbirth.. do you really want to add to it's stress? And secondly, if you're breasfeeding, then anything that you do to yourself has a chance (no matter how slight) of affecting your baby. Even if it's a small as maybe your body putting extra effort into healing the tattoo and a little less effort into the quality of your milk?
I'd stay away from getting a tattoo while breasfeeding for sure.. and even if not breastfeeding. I'd still wait 6 months after giving birth.
Getting a tattoo from a competent tattooist who always puts his clients wellbeing first is a safe procedure. Any decent tattooist will always tell you that the best time to get a tattoo is when you're feeling absolutely 100% fit and well. It'll minimise the trauma your body suffers and speed up healing time. My opinion is that if your body is doing anything extra, over and above normal function, (so healing after childbirth, breasfeeding, etc) then you should wait.
Maybe use the time while you wait to research the best tattooist in your area? Design your own tattoo? Find out a bit more about the tattoo industry so that your not giong into it blind? Just a though
I'm not trying to scare anyone and sorry for the ENORMOUS post.. but getting a tattoo should require a lot of thought and at least a bit of knowledge.. especially now that you have a little person depending on you

xxx