Dopamine is a neurotransmitter of the catecholamine and phenethylamine families that plays a number of important roles in the human brain and body. 

Dope is a slang word designating any illicit drug, and in particular marijuana or heroin.

So, what is the relationship?

Well, from a semantic standpoint, there is a strong connection as the dopamine substance is involved in drug addiction mechanisms within the brain. 

However, the similarity between the words is pure coincidence.

The word dopamine is actually a contraction of 3,4-dihydroxyphenethylamine, whereas dope comes from Dutch doop, which means sauce (and sauce became drug, etc.).

For those who speak French, I heard a French university teacher saying the other day that when he told his students about dihydroxyphenethylamine, they were running away, so he now says that it’s short for Désir Obligé de Passage à l’Acte (Obligatory Desire to Act), and the students remember immediately!