Second would be cohesiveness. The best concepts are built around one thing or another, and allude to that/those thing/s in a number of ways. For example, lets start off with something simple, such as... a description.
Our character concept is best described as a rough, gruff, bar-fightin' brawler. A man with two swords, and an abundance of muscle.
Okay, so our character is big, beefy, melee and so far, entirely physical. This eliminates magic-based, or ranged abilities, such as... Swinging a sword/both swords with great force, an shooting off a firebolt or something. There's no mention of any magical elements attributed to either him or his weapons, so lets make sure his abilities don't scale with AP.
Now, lets remember his description. He is described as a man who often picks fights in or around bars, taverns or anything of the like. He'll more than likely be the type to kick over a table at some point- possibly during the fight. He uses the terrain to his advantage. So what can we do with that? Hm... Lets give him the ability to blind someone with a handful of dirt or sand.
Keeping with that theme, the rest of his abilities should be somehow related to his rough-and-tumble fighting style. Things that would promote a playstyle that involves wading directly into battle.
M'kay, so lets say the kit is finished. What next? Well, since we started with a description and a kit, we should come up with his lore, or atleast his place of origin. Of course, you can start with any one of these things and build off of that, instead of doing things in this order.
Now, consider the current trends in League Champion releases, or lore events, so as to avoid jumping on a bandwagon. If we were having this discussion during the Freljord event, which is relatively lore-heavy, releasing this character as a Freljordian would make it seem out of place, what with the 'three-way war' going on. In that case, wait until after the event if you want to give him a Freljordian origin.
If not, lets keep our character in mind. So far, when you think of this guy, what image pops up in your mind? For me, I see a Shuriman. A large man with a pair of scimitars, no shirt, poofy white pants, metal boots and gloves, and some other bits and pieces here and there. Possibly with a turban. Now, this character has an appearance and a personality.
I won't go into detail, but lets give him a backstory. To summarize, a fitting lore might involve this bar-brawling swordsman hoping to test his skills in the League of Legends, or sate his hunger for battle.
So now, the basics are done. We've got a character that is easily defined, with a distinct personality, shown in both their kit and backstory, and possibly their voice clips. I don't often see other concept creators do this, but at this point, I come up with their base stats, stats per level, base values for abilities, and scaling. Done right, a simple bit of number juggling on their stats per level, and ability base values per level/scaling per level can determine what kind of champion we have.
For example, if this champion has a direct damage ability on a 22 second cooldown, and every level put into this ability lowers the cooldown by 2 seconds, while granting appropriate additional damage(+25/50/100/150), they would most likely have a lategame focus. However, if their base stats are already high from level one, and have mediocre level scaling, they'd most likely be an early/midgame champion.
Basically, as a concept, your character can be as under/overpowered as you want, so long as the core elements are there, and they make sense. Ingame, numbers and features will be dialed back, or dialed forward, depending on the state of the character, but its still a good idea to pre-balance their kit. The main point, again, is that all aspects of the character make sense. A gun-toting assassin shouldn't fight solely with their fists ingame. A wizard shouldn't have an AD scaling uppercut ability unless it is explicitly stated in their lore that they're either some sort of 'spellblade', or practice some form of 'full contact magic'.