Remember which types of tiles you have already touched so you can aim to touch the ones you haven't touched yet. Just don't assume that ones like the forwards and backwards conveyors are the same, or you may not end up touching all of the different types.
Any character will work for this achievement. However, the Runner, Lizard, and Student may work best as their slow speed but good maneuverability gives you the most time to complete the achievement. The Student is best because she can jump higher than any other character. If you need to get height quickly, jump and immediately disable gravity.
The Achievement Bar is very helpful here, as you can see the position increase as you jump on a new tile type, and stay the same when you jump on a similar type, which tells you that you have already jumped on that tile type. Create some mental notes and jump through the level.