Select Skirmish mode and progress to the first Boss fight. The Bosses will appear with other enemy fighters. Drain the enemies' energy almost to zero, then build a combo of five kills before killing the Boss with several shots.
Easy "Combo Master" achievement
Play a random Sky +1 Insane battle chain. Use homing missiles and complete the level. successfully or it won't count. Because it is a battle chain, just get more points than your opponent. Dying is acceptable.
Easy "Elite!" achievement
Take off all of you non-elite weapons and leave only a single elite on your wing.
Easy "I Am Superior" achievement
This can be done solo. Add one player locally at the top right corner and select versus mode. There should be two players now available at versus menu, allowing you to start a match against yourself.
Easy "Seeing Red" achievement
Purchase and equip the Power Grab super weapon. Reach the Boss in Water Wars +2 under the easiest difficulty setting. Wait for a red Shadow enemy to spawn, then use the Power Grab super weapon.