One great way I found to check if a champion is broken or not.
Remember this? http://na.leagueoflegends.com/en/news/game-updates/gameplay/dev-blog-classes-subclasses
Basically, I looked at it again recently and I realised that most unbalanced champions usually can be put in two or more categories of subclasses which make up for each others weaknesses.
Here are some examples. Keep in mind that this method only shows why a champion's kit is broken and unbalanced and it doesn't necessarily show if a champion is overpowered or not.
He is an artillery mage/battle mage hybrid and this is where all his problems stem from.
The big weakness every artillery should have is that once you close in on them, they're screwed. They make up for that by of course, having really high range that allows them to constantly poke you from a safe position.
Yet Azir also has the insane mobility and close range fighting capabilities of a battle mage, which means that if you close in on him, he will just push you away with his ultimate or shuffle away with his Q-E combination. Usually battle mages are balanced around the fact that they have really low ability ranges so that they are allowed to have a lot of defensive tools. Yet Azir has so much insane long-ranged poke DPS that this makes him broken.
As you can see, every weakness one of the two classes Azir is a part of has is non-existent because the respective other class makes them so. This is why Azir is broken and unbalanced.
He would have to either lose some of his defensive capabilities so that he can work as an artilery mage or some of his ability ranges to work as a battle mage but he can't have both.
He is a vanguard/warden hybrid and this is where all his problems stem from.
Every vanguard has top-notch engaging capabilities and they are usually the champions which start a fight. They make up for that fact with their core weaknesses thet every vanguard should have - that once they've engaged they have a really hard time getting out of a sticky situation or fight and they also are really bad at peeling their teammates.
Yet Gragas can do exactly that. His E, Q and ultimate also serve as defensive tools to disengage and peel for his teammates which a vanguard just shouldn't have and which effectively also makes him a warden.
Now Wardens usually have the weakness of not being able to engage a fight or make picks to start something to allow their teammates to follow up which means that they rely on them to do that for him. But since Gragas also can do that and is a vanguard too that isn't a problem for him.
This makes it so that Gragas doesn't have any weaknesses in his kit which makes him inherently broken and unbalanced. He would either need to get less defensive capabilities so that he can't just engage without consequences or he would need to have less offensive capabilities so that he is able to disengage and peel and be a very safe pick but not be able to make picks or start engages to make up for it.
Probably the most insane of them all, he combines the aspects of an artillery mage, an assassin and a skirmisher all in one champion.
His E-Q combo in his ranged form makes him an artillery mage. This gives him great long-ranged poke from a save position. Usually artillery mages don't have a lot of defensive capabilities to make up for that but . . .
. . . he also has his E-knockback and multiple speed boosts in his kit which are the defensive tools that let him excel as an assassin, which are able to burst one squishy target and get out quickly after they've done that with the defensive tools I just mentioned. Usually assassins have no poke though, and they also don't have enough DPS to deal with tankier targets but . . .
. . . the fact that Jayce has super low cooldowns on his abilities and because he has mixed damage with some percentage damage spread in there means that he also excels at being a skirmisher. Skirmishers have a lot of DPS but are usually lacking burst and poke to make up for that fact. But since he actually has poke damage as I just mentioned and burst too this doesn't really matter to him.
This makes him super unbalanced and broken and he would need a full rework to change something about that. If he would become a skirmisher he would need to get rid of the Q-E poke and some of his super high ability ranges on top of some of the burst he is able to put out. If he would become an assassin he would need to get rid of the Q-E poke too and of all the DPS and percentage damage he has in his kit. And lastly, I think making him an artillery mage is practically impossible because he has a melee form so he's pretty much left with the choice of either becoming a skirmisher or an assassin whenever he's going to get reworked.
Else he will just stay in his broken forms that inhabits all three subclasses that eliminate each others weaknesses.
So basically, check if you can find similar champions that inhabit two or more subclasses that just don't go well together and voila, you found yourself a broken unbalanced champion. These kind of champions have kits that just can't work in League of Legends and they will most likely end up either dominating the Meta or being sub-par.
