Steredenn: Binary Stars

After the Binary stars update, you are required to kill the extra 6 bosses + a few more variants of the original bosses (21 total).

Here's how you unlock the extra 6 bosses to the arena:

An Old Friend:

You need to quickly collect a golden coin that randomly appears after killing an enemy. After collecting it, the boss spawns on the next stage and gets added to the arena.

The Red Baron:

This is the most difficult boss to unlock. You'll have the clear the first 3 stages of the game without taking a single hit. So you'll have to kill the bomber (4 variants), frigate (3 variants), and destroyer (2 variants). After killing the destroyer without taking any damage so far in the game, the boss spawns on the next stage and gets added to the arena.

Garbage Collector:

You'll have to let 2 consecutive cargo ships escape the screen without destroying them. These are the ships that drop weapons. Then the boss spawns on the next stage and gets added to the arena.

Alien Threat:

Sometimes after a level on the upgrade screen there will be an Ancient Power upgrade. After you pick it up, you get a power, that deals damage to the ship when you get close to them. Some time later, a blue ship will spawn and fly across the screen. Get close to this ship and let Ancient Power kill it, then the boss spawns on the next stage and gets added to the arena.

Asteroworm:

Sometimes there are asteroids with tentacles during a level. Once you destroy enough of these, the boss will spawn on the next stage and gets added to the arena.

Super Chabal

You need to kill enough regular Chabals (little ships that attack by charging you and trying to deal damage by a direct contact) on a single level, then the boss spawns on the next stage and gets added to the arena. These little ships are the only ones that don't try to shoot at you, you should recognize them pretty easily.

Ok. So the original guide is correct (mostly), but I figured I would add some general tips/a cool trick.

You don't need to have beaten Nemesis (the true last boss) to completely unlock arena mode. Beating the Mothership unlocked the last Arena fight for me (as of now 9/23, I have not completed the game, but have completed Arena mode).

When selecting each boss, you do not need to complete both a one weapon and two weapon run. I did one weapon wins for the first 8-10 bosses, and 2 weapons for the last ones. There are 13 fights to complete. Here are some tips/thoughts for each:

Bomber, Frigate, and Destroyer - These 3 should not present problems. If you have unlocked the other bosses, you will become quite proficient beating these bosses without any upgrades. The Homing Missile works well enough for all of these 9 bosses. The only one I wouldn't use it is the Destroyer variant with its own Homing Missile. For that one, consider using a bullet weapon.

Cruiser - I found the homing missile and reflector worked well. If you'd prefer a different attack weapon, bullet or otherwise, feel free to replace the missile. Basic Strategies:

-For the half moon arcs with the blue/pink bullet attacks, use the reflector to clear some space around you to move around. Makes this section easy
-For the blue streams with the occasional pink big shot, stick to the right side of the screen either above or below the cruiser. The blue will strictly be on the left side of the screen while the pink shots will be the only thing you need to dodge
-When the cruiser stops firing, it moves to bottom of the screen and sweeps with the green shots that split into a bunch of blue shots. You can actually pass right below the cruiser and not get hit by him. Then all you need to do is dodge the blue shots. Consider using the reflector is you are having problems
-For the laser V, stick to the center and then move as soon as one of them disengages
-When it's the final part as the main cannons blow up, just keep your distance so you give yourself time to see where the shot is going. It appears random to start and the path becomes clear as soon as it exits his general area.

Carrier - Supergun is all I really need for this fight. It will take a bit, but this is the safest weapon to win this battle. The spread allows you to kill the fighters quickly before they can get behind you or shoot you themselves. If you want a second weapon, consider one of the ultimate laser/energy weapons. They can cut through multiple blocks quickly. Basic strategy:

-Just fire the supergun. Even as it places the blocks, keep firing. This will lay down a good cover to destroy the blocks while also destroying the fighters. The best thing to do is not allow a lot of them to build up. They can easily overwhelm you and kill you.
-When the bottom right of the ship opens up the rear door, two laser enemies will come out. Each will fire their laser once or twice. Just stay out of their way and you should be ok.
-The only truly annoying part is the firing of the horizontal bullet that creates vertical lines. Make sure you are not line with the ship so you don't get hit with the primary shot. Then just avoid the lines that those shots create. Continue firing the supergun to kill any fighters coming out.

Battleship - Supergun and Reflector - A different primary firing weapon works too, I only prefer this weapon since it always has some bullets hitting the enemy. I rarely used it though since the bullet hell is full blast here. Basic Strategy:

-Deflect Bullets!!! The boss only moves on to the next part when it has taken sufficient damage. In the opening sections dodge while selectively deflecting a few shots each time. The amount of time to reflect is very short (especially without the upgrade), so you only want to do this in short bursts. This took some time to get used to, so don't worry if you overuse it your first time or two. You'll get the timing down eventually.
-The really annoying part of this fight is when there is a mix of bullets and missiles. The battleship missiles are not nearly as good as the Destroyers, so you really only need to slightly dodge them rather than completely destroy them.
-The one section you need to fire a weapon to get him to move on is when he does the spiral shot in and out. There really are not enough bullets on screen to deal sufficient damage without it taking forever. Therefore, start with the reflector and move out the shots around you and do the same when he draws them back in. You could probably get off a few shots at this point. After four of these, he will engage in a longer break where you could focus on shooting a bit more.

Mothership and Nemesis - Supergun and Reflecter (or some other bullet/shooty weapon instead of the supergun). While killing the Mothership unlocks this fight, actually completing it requires beating the mothership and the nemesis. If you are this far and used my strategy for the battleship, you should be pretty good and handling the reflector. Basic Strategy:

-Deflect Bullets!!! No two ways about it, this is the best way to survive. This fight is more a battle of attrition than actual doing the damage yourself. Managing the power meter for the reflector is going to take patience.
-The opening set you arc out multiple bullets. Deflect the one in your direction to keep space open. After so many stack up, hit some of the ones in front of you into him.
-The next pattern is a huge spiral bullets out at you. Simply deflect and then move back. only get enough out of the way to keep yourself safe.
-The only really bad part is when you get waves of bullets with the lasers. The hard part is paying attention to the guns. There are 3 cannons that fire them with a red crystal on them. That's where they are. Just stay out of those lanes and you should be able to deflect a few bullets at a time to do some damage.
-You will be towards the end when the jaw mechanism opens. Deflect the shot it fires and then some of the other small ones. After a few shots, the mouth will close and you get an opportunity to fire substantial bullets.

Eventually, it will die and the Nemesis fight begins. This fight is "short" by comparison. Therefore, there isn't much to say:

-There will be some small bullet waves/flowers you can deflect, but you could also fire the supergun to destroy his shields. He maintains a similar attack pattern throughout, so your strategy will not be different during the fight either.

Upon completing all 13 arena fights once, you will get the achievement.

The Arena allows you to choose your own weapons (one or two) and battle each of the individual bosses throughout the game.

The only kicker here is that you cannot battle a boss in the Arena until you destroy them in a normal playthrough.

Once you have them all unlocked, simply choose your favorite weapons and destroy them again.