Puedes decirlo de las dos maneras, es correcto en ambas formas.Un saludo!

That's exactly the same. But the first sentence is ambiguous, because it could means that you want to make a tattoo on your bosy by yourself, and not that another person do it on you. But equally, both ways are the same and we, the native spanish speakers, interprete those phrases as the same thing (usually the first one is the most used). And although "quiero hacerme un tatuaje" is ambiguous, it takes the ambiguous meaning depending on a specific context. I hope I'd helped you! :D

Oh. I didn't know this.thank you very much indeed.

You can say it both ways, but they don't have the same meaning. If you say "quiero hacerme un tatuaje" means that you want a tattoo and you 1) may want to do it yourself, or 2) you may want someone else to do it for you. If you say "quiero que me hagan un tatuaje" then it can only mean that you want someone else to do it.