This one was surprisingly tough. You need to earn at least 10,000,000 points to reach Captain rank and the points are not cumulative from game to game. I did it with the Bomber class because it is the fastest ship and has a powerful weapon. The Bomber is also the weakest class and can only take 2-3 hits once the enemies become red. Be patient and play conservatively. Stick to the edges of the maps because many enemies tend to gather in the middle. Use your crappy powerups immediately to try and get Invincibility, and save it until you're about to die. There is one level where a ton of space pirate ships come and drop huge amounts of point boxes; there are also Temporal Distortion powerups on this level you can use to gather a lot of point boxes. More importantly than trying to get kills and points, stay alive. Patience really is key.
As a possible alternate method, I just used a flagship myself. I don't know if you get bonuses to your score with a bomber but by using a flagship and being accurate, I got an extra 5,405,847 points with my in-game score being 6,778,750. These added up and put me well over the 10,000,000 mark.