Champion's Road is a b***h. To be honnest, beside Peach and Rosalina, you'll be doing A LOT of trips to Star-5 (Super Block Land) to get Super Leaves.
First section: Octoombas.
With Luigi, Peach and Rosalina, it's possible to get accross the first large gap as long as you get a running jump at the beginning. Mario and Toad need a Super Leaf to do so. For the last part with the two moving blocks, it's possible to get across easily without much waiting. With Mario and Toad, you can long-jump your way across. With Luigi, he jumps high enough to not need to long-jump. Peach can hover across and Rosalina's double-jump does the trick. Predicting when the blocks will move is key in a good jump ahead. Also, those blocks moves FAST.
Second section: Fire Bros.
Luigi and Rosalina can jump hover a block high (Rosalina needs her double-jump however). The rest will have to jump around. Oh and... just don't care about the Fire Bros. If you keep moving, they won't hit you at all. Just keep in mind that they move away after two fireballs.
Third section: Small gaps & Chargin' Chucks.
Small hops are your friends. Small hop from section two onto a bar and jump over the first chuck. Note that, like in 2D Nintendo platformers, you can run across those small gaps. Then, jump on the safe platform. Jump on the bar that is attached to the wall and start running and dodge the first two Chucks as you deviate on the farther bars from the wall and jump over the two other chucks. Peach can simply ennemy-hop her way across since her hovering is handy for that.
Fourth Section: Bip-Blocks
The most difficult section of them all. You will be raging. Especially if you try not to use Super Leaves. I don't have magical recipes here, unfortunately. I personally wasn't able to get a reliable strategy for Mario and Luigi on this one. Getting a running start and getting on the blocks before they even appear (first blue set) is important as you can start running, jump on the other set (red set) and then on the third set, you need to run just a little bit and long-jump to safety.
The second part is tricky too. start jumping from the red block to the blue block above then to the red one beside and then to the blue one on top and then the higher red one and then jump across to the blue one and onto safety. The last red one is the difficult one. From the blue blocks, if you simply jump onto the next red block that is on the same level than the blue one, you are doomed unless you find a way to jump back down and try again. So, you have to jump on the higher one and then long-jump immediately to the blue one across a gap. I personally didn't rely on that strategy but it seems what Nintendo had in mind. As Luigi and Rosalina, with a Super leaf from the last higher red block, you can hover to the safety zone. With Peach... Peach as hovering. So you can technically take your time and wait to get your timing to jump across. Because, yes, the hardest part with Peach, specifically, is to get the right timing. You have to take in consideration that you have no momentum. Rosalina can save herself with a double-jump after doing a walljump if anything goes wrong but Peach can't if you miss your boat...
Third part is... it starts easy but when two of the same color blocks are next to each other, the strategy is to just keep jumping ahead. Peach and Rosalina saves face once again with hovering and double-jumping.
Section Five: Defeat the ennemies.
Three magikoopas. Just don't stay in place and avoid the corners. Any power-ups helps.
Section Six: Spiky Swings & fuzzies.
Pure platforming. The Pot Pirahna Plant helps tremendously with the fuzzies on your way. Again, Peach and Rosalina has an easy life with hovering and double-jumping. Just note that Rosalina loses her double-jump if you take the plant.
Section 7: Walljumping and ennemy-jumping.
The first part will be wall-jumping with Spiky Ant Troopers walking on the walls. It's possible to just wall-jump your way up in-between those Ants but... it's very tricky. If you have a Super Leaf, you should take the time to defeat all of them and just wall-jump your way up there. Rosalina can defeat them with her Star Spin.
The second part is jumping on Pirahna Creepers. Easy enough.
Section 8: Water block swimming
Just... just make sure you don't start to swim toward yourself (ie. away from the wall). When it reaches the right end of teh swimming part, mash-swim your way up, it's not tight but not loose either. Back on the left end, be sure to be already going down when heading for the clear pipe. Otherwise, you'll be likely taking a hit from the moving spiky block.
Section 9: Boost Pads & Ring Burners
Oh boy. The second hardest part in my opinion. So, it'd be best to get on the boost pad when the first Ring Burner's shock wave is already started moving toward you. It's important to get a timing here because it will decide where the other shock waves will be along the way. Super Leaf and Peach's hovering helps tremendlously dodging waves and clearing the gaps. The last part is the trickiest. You need to get keys that are at each corners and teh fifth one in the middle. Hovering will help you A LOT. Also, you have max momentum on this and it's easy to lose control. Be ready to take a hit or two (if you have a power up). Once done, get in the box.
Section 10: "Thank You!!" Clear pipes.
You can't die unless you miss the clear pipe entrance or suicide. That'll be stupid.
Section11: End Stage.
Well, enjoy the view. Be sure to get your VERY rewarding stamp and Gold Flag.
Congratulation on completing the course. To be honnest, if you are of sane mind, you'd prefer taking trips to Star-5 to get Super Leaves when doing this level other than Peach and Rosalina. Personally, I even got Super Leaves for Rosalina because, for me, hovering is more useful to double jumping.
My first time was with Peach without power-ups. Yep. That try I got it, I somehow been able to wall-jump in-between the Ants in Part 7.
It's very hard.
Hardest: Luigi ("leg struggle" then hovers with Super Leaf. It breaks timing)
Hard: Mario (No special abilities makes his run a pain)
A notch easier than Hard: Toad (Fast running helps the first part of Section 4)
The "Easier times": Peach and Rosalina.
Hope that helps!