Okay, you want to make an alt account for reasons: another shell, to go around incognito, or maybe you just want to have achieved  twice. The tutorial, known as Brink, can be a drag when you already know how to play. So here is a compilation of things you can do to make Brink go fast.
Brink has 7 rooms:
Make sure you just keep holding the right arrow. Depending on your keyboard, you will probably be able to jump while running, and you may be able to hit down+space while keeping the right arrow held down. If you can do the latter, let go of space as quick as you can, to keep yourself moving right while falling. Else, make sure that you minimize time spent not moving right.
Get that gun and shoot the wall. Then kill the minion to advance to the next room.
Method 1: There's actually a very small area of space to simply stand where your grenade will go over the wall and hit the enemies inside. Stand right past the ammo refill station and throw.
Method 2: Right after picking up the grenades, jump towards the ammo station and throw your grenade while you're over it.
Optional step: If you aimed wrong and feel you've wasted a lot of ammo on that, take a quick visit back to that ammo supply station. You'll need ammo for later. Else, just keep pressing forward.
Use down+space to fall through the first platform, or walk off the first and zigzag around the second. None of the enemies in this room can jump, and they are slow to attack. Jump over them and proceed onward.
It takes little ammo to kill the boss, and you can even reach him without aggravating him and starting the fight. Stand directly in line with the planet (you might have to stand just a little closer than that to reach, but the center should be sufficient). Then jump and throw your grenade at the peak of your jump. It takes two or three to defeat him. Use Shift+A to equip your shiny new gun and move on.
Just keep gunnin'. It helps to do little hops while shooting, so that you're clearing out the long line ahead of you, but still advancing. Keep an eye out for horizontal fireballs, since a knockback can slow you down a lot.
A grenade will reach the Watcher before your gun's range will. Jump off your initial step; if you throw from the end of that ceiling, a grenade should reach him. Once you've caused some damage, just sit and wait for the cutscene to play itself.
And voila! You're up and running in Eschaton, a sparkling new . Hope this helped, and thanks for reading!