I agree that the creation of achievements would need to be carefully thought out. That being said, I think league is the perfect place for them. The fact that league is a repeating scenario it has very minimal goals to achieve. For example I have been playing for 3 years now and have finally achieved all that I wanted. I have every champion, and have achieved gold two seasons in a row. Yes I know that there are more things to attain, all runes, higher rank, but those goals are either skill based or not very rewarding.
I think a way to set up achievements would be similar to a FPS. X kills, double triple ect. Assists, KDA, Smite's/steals, amount of teammates healed. All these stats are tracked per game, tracking the all time per account would add a huge dynamic for completionists. Another Idea is the way you need to do a certain number of things using different weapons in an FPS, this could be done using different champions as different weapons. This would also promote diversifying your champion pool. I mean think of the achievement "10 wins with every champion" this would improve game diversity, be a fun goal to go for, and take quite some time.
These achievements could unlock summoner icons which would I think appropriately reward players with a visible look what I have done.
I do think the longer term X kills with Y champion, or X healing done, or X champions cc'd, X damage taken, are better than the more competitive "skill cap" achievements such as X pentakill, or X no death games. Though these could be included they would just need to be tested to see if they interfere with the primary game goal, winning.