I did think about this for a while.
In order for this to work, it would have to be an ultimate ability (like you said) but have the damage scale with your missing health.
It would also have to make you invincible during your cast time, and when you activate it. Cast time is there for people to at least get a running start away from him. It would work either like Kog'Maw's passive, where he can run around, but then there's the problem of how fast he should run. The other option is to have him stay immobile.
I think if there is a cast time, then it would be somewhat ok if he could move after his cast time, but then continue to be invincible while he is running around. He would need to be targetable, so that if he gets hit with slows, then there is at least a chance to get away from him in case the person he has in his sights is super immobile.
Then of course there is the problem of team comps built around keeping this guy away during his kamikaze state. So maybe built in tenacity during that state? Or maybe just slow resistance.
Of course, this ability kind of changes how the gold is shared when he dies from his kamikaze. Does the kill goes to whomever last hit him? Does he die as an execute and then the enemies who were attacking him get assists? These are probably things to think about if it were to go to live in order to prevent hardcore feeding and trolling.
Then, final question, what kind of champion is this going to go on? Is he a fighter? A bruiser? A juggernaut? A tank? An assassin?
This ability should go on someone that can make an impact in team fights without an ultimate, and then when he does need to kamikaze, then he will leave the teamfight in glory because the player knows that he actually did something productive.
I love these cool ideas!