I am assuming the addiction here is in the context of strong drugs like meth, coke etc. and the literal meaning is similar to phrases like regular consumption, heavy consumption and heavily dependent.
For image of an addict, may be the character Bubbles from The Wire, fits the bill.
If that is the case, then in my opinion, Jesse is definitely an addict in Season 1 & 2, up to the drug overuse scenario which causes Jane's death. But after returning from the rehab, in season 3, his character definitely changes. From season 3, I don't think he is an addict.
He is seen having his moments of despair and aggression, where he does seem to be under the influence of some substances, but definitely does not seem like an addict from season 3.
It should be noted that lifestyle also plays a big role in the story of addicts.
In season 1 & 2, he is kind of a loner. He meets Jane, but she herself is an addict.
In season 3, he meets Andrea, who is a recovering addict, whom he meets in a group session. I think this is a pivotal point in Jesse's life. Since he is with Andrea till season 5, his lifestyle definitely changes and he is certainly not an addict anymore.
It's noteworthy, that from season 3, even Jesse's dressing style changes.