What was his problems? (SPOILER)

It's kinda the underpinning of the whole film that Jack just wasn't ok mentally. He 'had it all' but there was just an inability for him to be at peace and stable in the world he lived in.

We first see him on film immediately drinking after his concert, he does drugs throughout the film. He has an upturn when he meets and marries Ally. But it's hinted at throughout the film that he had a lot of demons from his childhood, like the fact that his father was a raging alcoholic who he idolized because he was the only parent around and he was supposed to, most likely, isolated away on a farm. Jack mentions however it was bobby he idolized really, but Bobby in Jacks early years was always out at gigs trying to make something of himself (jack father had him in his 60's, Bobby probably in his early 20's), given Bobby was like at least 20-30 years older then Jack. Jack mentioned that when Bobby had left once when he was 13 that's when he tried to kill himself, and that the ceiling fan he'd tried to hang himself via had sat there broken for months (probably when Bobby came home). It must of been pretty fucked up for Bobby as well to feel like he was basically being surrogate parent to his younger sibling.

As for why he had to commit suicide. The character, just couldn't handle life and his demons, likely knew that he was always going to be the way he was, some people will never change. Perhaps he knew he was on his way out career wise, Ally was on the up and up per how she was being marketed, and that the downward spiral of himself would only drag her down like a rock.

I do hate that they made Ally transform into a pop persona, the plot could have worked easily without the pop star stuff, but it's more relevant today and an easier vehicle for the conflict i guess (Jack being staunchly country rock while Ally could do whatever marketed her best, similar to Lana Del Rey and Lady Gaga herself)

He killed himself because the manager made him believe he was going to ruin Allie's carreer.

I think asking why he did it/what were his reasons displays , sorry, a certain naïveté about addiction and emotional trauma generally . As a pp suggests, he was a damaged person from way back , and in any case alcoholism is not about a simple choice to drink or not.
In addition, he believed he might ruin her life and he loved her too much to risk that. And finally, he was getting inoperably deafer at an increasing rate .

Why was Kurt Cobain's problem? Chris Cornell? Chester Bennington? I guess those with great artistic ability are prone to depression, or other mental disorders that predispose them to suicide

He killed himself because he was
1) going deaf. His own career was essentially over.
2) his alcoholism and addiction would never be controlled and the manager broke whatever confidence he had
3) he felt that he was holding back his wife’s career as she was cancelling tours to be with him

It’s tragic but he saw no light at the end of any of these tunnels.

Cause of death: too much of a good thing.

Just consider the real world in comparison and you'll see it's not uncommon..

When we see people who seem to have it all and still want to kill themselves it seems pretty obvious that having it all doesn't necessarily always make it better. Sometimes it makes it worse.

If you have it all and are still miserable or suicidal then it must feel like there is no hope..no escape. There's no solution.
Life is as good as it gets and you're still miserable in your own skin.

Of course, Jack had been slowly killing himself for many years anyway and the catalyst was believing he was sabotaging Ally's career..
In some ways it was an excuse to do what he'd wanted to do but he needed to believe he was doing it for the right reason and that she'd be better off without him.