The player level is the cumulative stage scores from all seven levels. To reach level 6, you need to reach a cumulative score of 70,000 with one of the characters.
You only have to reach level 6 with either Ares or Tarus, but not both. So pick a character you are more comfortable with and focus on improving your scores in each of the levels.
At the end of each level, you are graded on a few criteria that determine your overall score. By getting "gold scores" or perfects in each of these criteria, your overall score will increase. These criteria include:
-Number of Times hit by Boss
-Max Combo Achieved
Helpful tips going for the achievement:
1) Do not attempt this until you have first finished the game, found all the upgrades in each stage, and fully upgraded whichever character you are using. By finding all the upgrades, the "Upgrade Found" category at the end of the stage will always give you a "Perfect" score. Having a maxed out character will help you do more damage and easily steamroll the level more quickly.
2) Since you need an overall score of 70,000, you need to be averaging at least 10,000 points per stage. To have a better shot at doing this, up your difficulty from Normal to Hardcore. Enemies will have more life in Hardcore, but you should hardly notice the difference with a fully maxed out character.
3) Use the fully upgraded Wave Emitter Gun to destroy as much as possible and rush through the stage. Don't worry too much about taking damage-- damage received during the stage does not penalize your score, only damage taken against the boss. Try to string a high combo on the way, but don't worry about it too much-- Most of your combo will likely come from the boss if you can destroy it without getting hit
4) You can still get hit about 3-4 times from a boss and get a perfect rating for the stage. Make sure you are using a fully upgraded photon cannon against bosses, and you will destroy them in no time.
Playing through each stage on Hardcore, I was able to easily get an "SS" rating playing as a fully upgraded Ares. I averaged a bit over 10k on each stage, and scored over 11k on the first stage. This enabled me to only get 8.5k on the last stage and still end up with 71k overall for the achievement. Once you get the pattern of the bosses down (which you likely will going for the "Perfect" achievements), this cumulative score achievement shouldn't be an issue at all.