What is the fastest way to level an account?

Honestly levelling in League of Legends can be tedious and without buying XP boosts could take you a very long time. The fastest ways to level 30 are covered in this blog post but the general key points are:

Experience boosts (duration): Duration experience boosts give you bonus exp for a set number of games, regardless of if you win or lose. The best value boost is for 7 days and costs 520RP, or around $5. Depending on how often you play, you might need several of these before you reach level 30. You should be able to reach level 30 within a month using these boosts.

Experience boosts (wins): These experience boosts give bonus exp every time you win a game - but will give you nothing if you lose. The best value boost is for 50 wins and costs 980RP, or around $10. You'll need at least 2 of these to reach level 30. They can also be combined with duration boosts for bigger exp gains.

Playing normal games: Normal PvP games give much better experience rates than bot games. However, you will likely win around 50% of your normal games on your way to level 30, whereas you will probably win closer to 100% of your bot games. Without boosts it can take upwards of 250 normal games to reach the required experience for level 30 - that could be up to 250 hours of playtime! Definitely not quick if you aren't using experience boosts.