Dermatologists have been evaluating patients with tattoos for decades, and they have never found an increased prevalence of skin cancer in those individuals. The same is true for patients who have already had melanoma or another form of skin cancer — the inks used in tattoos have not been shown to increase their risk of recurrence. People who have had skin cancer are always at higher risk of developing future skin cancers, but tattoos do not increase that risk.
However, it's never a good idea to have a tattoo placed too close to (or over) a mole. Changes occurring in a mole — to its symmetry, border, color, size, shape or texture — are potentially key warning signs that the lesion may be evolving into a melanoma or another type of skin cancer. It's important that all moles are left completely visible, you could have a risk of delaying detection. It is also not advisable to tattoo over a melanoma scar. It's important to be able to look at the scar in case of recurrence. Tattooing will cover the area and potentially allow a recurrence to be missed.
When skin cancer is discovered and treated early, it is usually curable. More advanced skin cancers become harder to treat and may become disfiguring or even deadly if allowed to spreading. This is why anything that delays detection can be extremely dangerous. If you get a tattoo, make sure it is placed far from any mole or other spot that is changing or that concerns you. This is especially important for people who have multiple moles or atypical mole syndrome, since they are at increased risk of developing melanoma.