Secret Levels can be found in normal levels that have an alternate red and black portal. They usually require backtracking but are easy to spot. Once you've entered a secret level you can access it through the level select.
The climb before the final boss can be skipped once you've done it once, so don't be afraid to take a break/quit once you've made it to the boss.
Deathless runs can be done in any level order, and you can quick restart/quit to menu before dying to preserve your run.
For the MP achievement It Feels Good you can set the win condition to be First to 3 in Sunrise Mountains. Tutorial world cannot be selected sadly.
For PB Hype you can do an individual world speedrun of the tutorial world.
For Following Directions you have to do the 2nd tutorial level "Double Jouble" in one big jump without double jumping.