Challenge levels are accessed via the main menu and there are a total of ten of them that you need to complete for this achievement. A challenge level is unlocked for every 10 shinies you collect in the story mode, so you'll need to collect all 100 shinies to be able to access all ten challenge levels for this achievement.
Challenge levels are the most difficult levels in the game, but thankfully there are no restrictions or requirements, such as shinies, par time, etc. You only need to get to the end of each level. Easier said than done though. Some are pretty difficult. Expect to die a lot. That being said, they are still all manageable with some practice, and like story levels, with muscle memory developing. Just remember to take your time. There is no need to rush.
Here is some more information about each level:
Level 1 - this level is pretty straightforward in terms of how to complete it. You'll just have to have good control over your jump height to make it through all the obstacles, but there's nothing overly complicated here.
Level 2 - this one is straightforward as well. You have to ride the platform down to the bottom, fall off the edge of it and change gravity so you're on the bottom of it, then ride it back to the top and drop off the edge to the endpoint. I found it easiest to stay on the far right side of the platform initially, jump left over the first single saw and pull back to the right. Then jump from right to left over the big saw. Stand in the center of the platform and jump to avoid the double saws. Repeat on the way back up.
Level 3 - the smoke blocks in this level set up your path for you. The one important aspect though is that when you get to the first platform with the two saws, it will seem like the wall off to the left is unnecessary and you can just skip it and go straight up the nearby walls. This messes up your timing when you get to the four saws on the next platform and you won't be able to jump over them. So when you get to the two-saw platform, jump to the left wall, touching the ground as little as possible, wall jump off the left wall back to the right and then proceed up. This will sync the saws so all four are coming left while you're going right, allowing you to jump over all of them in one jump.
Level 4 - this level is a pain because of how easy it is for a homing bullet to catch you off guard. Head right and jump over the lower bullet and wall jump onto the smoke blocks to clear them, then fall down and through. There is a safe spot right next to the launcher shooting up, where the bullets coming down can't hit you and the launchers in the previous room can't hit you. You can get your bearings here. As soon as the launcher fires up, wall jump up to the smoke blocks and hit the wall twice to make most of them disappear, then quickly jump left to avoid the bullet coming up, slide down the left wall and then jump through the opening. The important part here is when you jump through the opening, wall jump up the back of the wall so the homing bullets can't hit you; don't go down to the floor. Jump up over the wall and over both launchers to the smoke blocks, wall jump up higher to avoid the bullets, then fall back down into the space. This is another safe zone. Get rid of the smoke blocks and now just one obstacle left - the fast-firing bullets in the next room. I found it best to jump over 2 bullets with each jump, then hop over the launcher and wall jump up to the portal.
Level 5 - the first part of this level is straightforward. Then you'll come to a vertical wall with big saws at the top and bottom. I'm not sure how the game intended you to get past these saws, but I found it easiest to wall jump off the first wall to the left to the higher wall, wall jump off this higher wall to the right and land on top of the first wall you jumped off of. It's a very small amount of space you have to land on, and you'll probably miss it the first few tries, but once you get it, it's easy to land on it consistently. From there you can simply jump over the saws. For this next section, jump over the clockwise saws and go under the counter-clockwise saws. This last part with the very fast saw is going to look intimidating. Timing is crucial here because of how fast it moves. What you have to do is jump to the first tiny platform, then to the second tiny platform, then jump to the wall on the right to perform a wall jump. At this point the saw should pass you. Wall jump left onto the top platform and then you'll have to quickly jump again to avoid the saw. Jump right and finish the level. That first jump onto the first tiny platform is what you have to make sure to do very early. Jump when the saw is down at the bottom so you're landing on the tiny platform immediately after the saw passes it, otherwise you won't have enough time to get out of the way.
Level 6 - this one is an auto-scroller so there isn't a whole lot to say. You'll have to just get used to when to jump. The hardest part is the saw blades near the end. After clinging to the wall and wall jumping back to the left platform, do a small jump to avoid the highest blade, jumping left to right, then do a high jump right to left to jump over both blades. Once you're under them, jump across to the right platform and finish the level.
Level 7 - there are two main problematic areas of this level. The first part is easy, you just have to get all three porcupines in the beginning to chase you to the left side of the second room, then wall jump over the wall to the right and fall and let them chase you as you run right, hopping over the fourth one onto the wall jump and back to the left to the safe spot. Now the first problematic part: this jump is impossible if you try to do it from the last small platform over all the spikes. There is no way to make it from being stationary. The way to do it is from the safe spot, jump up the wall on the left, wall jump right to the first platform, and from here you need to be holding right the rest of the way. So after wall jumping to the first platform, do a max height jump to the second platform, and then RUN OFF the second platform to the lower small platform. Now you just need to jump at the very edge of this small platform to make it over the spikes. It'll take a few tries to get used to but then it's easy to do it consistently. As soon as you get over the spikes, jump over the first porcupine and then the second. Now you have to do some precise wall jumping. When you get to the top left, let yourself slide down the wall enough so you have head space to jump over the spike at the top of the wall. This is the second problematic area: knowing what to do when you jump over that spike, because there are three porcupines waiting for you in the next area. When you jump over the spike, hold right the entire time and you should be able to jump over the first two porcupines. Quickly jump right over the third, then turn around and jump left over all three in one jump, jump on the wall and wall jump back into the portal.
Level 8 - if you haven't mastered wall jumping yet, you will by the time you're done with this level, as it's non-stop wall jumping through obstacles. There isn't a whole lot to explain since it's basically all going to come down to skill and being able to manage the jumps. In the first section, I found it easiest to jump between the spikes by being on the bottom pillar and being right under the bottom spike, then jumping up and around and through, instead of jumping through the gap from the opposite wall. You'll reach a platform where you can take a breather, and then even tougher wall jumping. Once you go down, through, and then up to the top of the first section, jump right over the saw and then I found it easiest to align yourself to fall straight down the middle and then quickly cling to the left wall before the spikes. Jump through the gap and back up to the top, and then again do the same thing when you jump right - let yourself fall down the center, then cling to the left wall and slide down to the bottom. The last section of wall jumping back and forth between the spikes is the most precise part. You have to be right under them before jumping to make sure you'll go high enough to get over the spikes on the opposite wall.
Level 9 - this level is thankfully much more straightforward and simpler than the previous and next level. Start by immediately wall jumping off the wall in front of you back to the wall above you, and then through the gap back onto the platform. As soon as you're clear of the wall, jump into the opening and do small jumps between the spikes to dodge them. Wall jump a single time up to the top and land over the first spike. Jump up to the top and drop back onto your platform, making sure to duck under the next wall. Once you get to the saw blades, the important part here is knowing when to jump to the next platform. It's going to feel intuitive to jump when your platform is moving right but you have to do the opposite. The saw blade should be going up to its low point when you're moving left. This is the time to jump right over the blade at max height, which will give enough time for the next platform to come back to the left for you to land on. Repeat for the next one and then finish up the level.
Level 10 - the first section is the most difficult section. It's got the most precise wall jumps in the game. Some may seem impossible to reach high enough, but the trick here is that when you jump into a wall and cling to it, you sort of slide up it slightly. Make sure you let yourself slide up the wall before jumping to the next wall to get yourself maximum height. Getting past the big spike requires you to slide up the opposite wall slightly, just enough to get past it to jump above it. Once you're above the big spike, you'll need to jump twice on this wall to get higher to make it left. Once at the top on the left, let yourself slide down the wall as far as possible to give yourself enough head space to jump right over the spike without hitting the ceiling. Up at the top, you've got to make some very short jumps between spikes. You'll need to have some right-ward momentum but not running or you'll hit the next spikes. There are also two glowing pads here that change your gravity. It is very important to run into these while running right or they'll toss you right onto a spike. Once you make it to the right side, you now have to go down. It is possible to jump and just fall straight down the center, but it's very risky since there are spikes on the floor waiting for you. Better to be patient and take your time, slowly sliding down each wall and then lightly jumping to the next one to work your way down. Once you get to the bottom and into the portal, congratulations on finishing these very difficult levels!