After the cutscene, you'll be given the task to chase after a small critter. It doesn't matter whether you get close to it, past it or not, just run to the right.
When the enemy soldiers are dead, run up to the gate and you'll be introduced to a new concept of using your new pet "Skiffy" to gather items and get past certain areas in the game. Press (E) when this symbol appears.
Run up ahead after the gate and onto the cliffs.
You'll come to another "Skiffy" checkpoint. Press (E) to lower down a climbable object.
Once it's down, get up on the large telt-roof and head right.
When you are up, press Kit up against the wall to your left and climb upwards.
When at the top, jump off the wall to the platform ahead.
After that, onto the next platform and keep going right.
A general rule in this level is to keep going up up and up. Look for arrows if you get confused at some point.
At the wall ahead, there will be traps that will within a certain cycle splurt fire out. Watch their patterns and avoid getting caught in it or you will fall off the wall.
Once up, head right and keep going until another Skiffy scene is available.
Get up on the lowered platform and push onwards.
The platforms will alternate up and down. Time your jumps to reach the far away roof ahead to the left.
If you drop down at the edge of the cliff, there will be a "treasure chest". It holds currency for you to purchase suits and weapons with later. Nothing too important.
You will see quite a few of the treasure chests, being hinted at you often behind walls and other contraptions. They are rather worthless since money becomes abundant later on. Ignore them if you want and head up.
When you reach the relative "top" of the map, jump off, following the currency in the air.
You'll run into another "locked combat" sequence where you will have to kill all the soldiers to proceed through the wall.
It's easiest to just camp the reinforcements after taking care of the "strong" mobs.
After the battle, you'll reach an elevator. Get on it and press (E) to lower yourself down, ending the level.