Level 30 is the maximum level you can reach with one character. For this version of the achievement you must select to be a Hero and your mentor to be 'Wonder Woman' when creating a character.
The quickest way to get to level 30 from what I've discovered so far is just play through the main storyline missions and nothing else. Don't do any side missions, or explore missions from Booster Gold, or Races or Arena's or anything like that, just follow the main story missions. This is because once you finish the last story mission you will already be around level 26-28 and on completion you receive enough XP to automatically upgrade you to lvl 30 (See video), thus earning you the achievement.
Now on your first character you'll probably find it took you ages to get to level 30, but by the time you know the story and where you are going you will be aiming to finish a character in 10-15 hours. This is dependent on how effectively you can use fast travel to get to where you want quicker and your skill level. Dying isn't the end of the world because in most cases you can just travel back to where you died and carry on from where you left off, but it is annoying and a waste of time.
To save some time (30 mins to an hour) when in a safe house and you just completed a mission and you have received a new one, just quickly check your journal and the mission details to see where you are needed to go. I say this because if it's another safe house you can fast travel from any safe point to safe point using the teleporter and it will save you time as you don't have to travel half way across the map.
Note: When you get to level 9 you can spend a skill point and unlock the ability to use super speed. (Hold the
while in speed/flight mode to activate)
Note 2: You get a free XP booster near the start of the game, this is actually useful because you can make your character stronger in shorter time to help you get through missions faster. Use it when you're around level 11-15 as missions come in thick and fast and you get a lot of XP for doing them.
All in all the missions aren't very challenging and by the second play-through you'll probably only die a handful of times. The only hard bit of the achievement is fighting the repetitiveness of doing the same campaign 6 times. At least switching between hero and villain does mix it up a little.
Also thanks to PhantomSiren
for helping me with info to compile this solution.