Another easy one, should be one of your first trophies as well. Either go to "Battle", "Quick Match" or "Private Match" and play.

The max level is 45, which can take you anywhere from about 30 to 60 matches (a win would earn me a level and a half to two levels, while a loss usually got me half a level to a level) or about 10 to 20 hours, greatly depending on skill or if you're boosting.

Note: You can press random to select a random character that will reward you with an extra 1000 Solar (experience) at the end of the match. Not the character you want? No problem! Just press and try again. You'll still get the 1000 Solar.

This one might give you some trouble. Just memorize the pattern and try, try again. Especially in the beginning where you're likely to be dying a lot. Or simply try it out in practice where you can die as much as you want.

Here's another video uploaded by benjamin1802 to help you out:

This can only be done in Ribbit IV, which is the first level you unlock in practice mode. You're looking for bushes in the middle of the level where you can hide from enemy Awesomenauts. Climb up there, then go either all the way right or left and it should be there.
This can be done alone or with friends, just make sure you deal the finishing blow. Also watch out as it's health regenerates over time.
If you're really having trouble finding it, just look at the trophy itself: That's what it looks like.

Here's a video uploaded by benjamin1802 to help you out:

You'll get this once you unlock Clunk (the big robot) at level 3, Voltar (the floating brain) at level 6, and Yuri (the Mojo Jojo (flying monkey) rip-off) at level 11.

A killing spree is when you kill 3 other Awesomenauts in a row without dying. You'll know when you've got it because not only does it tell you, but your characters theme song speeds up. You might need to boost to get this one. Just make sure you have help, and you teleport back to base (hold ) after every kill to heal up and possibly get an upgrade.

This one might be hard for some, but definitely do-able. As it says, you need to have more kills than anyone else on both teams (which is 5 other Awesomenauts). The only kills that count are from enemy Awesomenauts, not droids and/or creeps.

Note: If you're really having trouble with this then look for boosting partners here: Co-op Partners Thread (Read First Post!)

This one requires 3 controllers (DualShock, PC Logitech Pad, and the official Sony remote control for a Blu-Ray player all work, but not a Move controller). And if you happen to have a PS2 controller with a convert wire, it'll only take 2 controllers.
It can be done in "Battle", "Private Match" and "Practice". This can be quite tricky to do solo as the bots can actually be pretty difficult sometimes, so it's suggested you either actually play with 2 friends, or you get 1 or more friends on the opposing team to make it easier for you, e.g. letting you kill them for Solar, not helping their teammates, etc.

This one is easy and should be one of the first trophies you get. You don't even need to win. Just go to "Battle" or "Private Match", invite a friend or two and play. It'll unlock at the end of the match.

Note: It's possible to get this, "Party at Sky Vault" and "Yoooo JOE!" all at once. Simply go to "Private Match", get your two local friends to join your team and invite 1 to 3 friends off your PSN account to join the other team and win the game.

The six different characters (in order) are: Clunk, Froggy G, Yuri, Sheriff Lonestar, Leon Chameleon and Voltar.
You can go to Help & Options > Stats and look at the "By Class" section to see your win percentage. If any of them are at 0, you've obviously never won using that character.
Apparently you can do this either from start to finish or by joining late into a game. Just to be on the safe side, make sure you win a match where you were in it from beginning to end.

Note: This trophy has glitched for many people, requiring them to win more than one game with certain characters. If it doesn't unlock for you, don't worry, just win a few more games and you'll get it.

Another one that'll come in time, considering you can't win a game without destroying these. Turrets are the big guns seperating you from the enemy base. Each team starts off with either 3 or 4, although you only need to destroy 2 to get into the enemy base. If you really want to get this trophy quickly you should get your teammates to help you destroy the rest before you go after the enemies drillcore.

Easy trophy, it'll come in time. Using skills that attack multiple enemies (Clunk's Explosion, Froggy G's Splash Dash, Sheriff Lonestar's Dynamite) will help. Droids are the little robots that come out each team's base, and automatically goes after the enemies turrets. There are the little ones, the flying ones, and the super droids which carry rocket-launchers that spawn after you destroy an enemy turret. Creeps are the neutral enemies you can find in all levels (horned things in Ribbit IV, worms in Sorona, and robots in AI Station 404). Either way, you can check your progress by going to Help & Options > Stats and looking at the "Kills" section.