The "Extending Platform" level is in the Clocktower at the right hand side, near the boss door. There is a door just to the left of the boss door, then a door just to the left of that, it's that door.
Most of the gems in this level are fairly easy to collect without the rope, there is only one tricky bit that the rope is usually needed for.
You are supposed to wind the rope round the cog to move the platform across to the left so you can use the platform to get up to the gems. But obviously for this achievement you can't do that.
Instead you just need to do a tricky jump where you turn around in mid air. You can't get up, just by standing as far to the left as you can then jumping up, because then you are too low down and can't jump high enough. You still need to jump off the platform that you would normally move but without moving it. Since there is no gap above where the platform is though, you need to do it at a run.
So, the move is, run along the platform and before you fall off the left of it jump and then turn right in mid air to land on the platform above.
This will take a bit of practice. You will probably push the jump button too late the first few times while thinking you pushed it at the correct time, and that this move is impossible. But it's not.
You might want to collect all the other gems first before jumping up so you don't have to get back down, but I recommend not doing that. Just go for the jump as soon as you start. It's actually not very difficult to get back down if you do make it up.