Note: Since you can only use the Bunny for this achievement, it will be the only suggested character.
The only way to complete this achievement is to use the Bunny. Here's how to do it: Reach 500m, then hold down the up arrow, w, or the space bar for another 500m.
Tips: Try to jump to the side of the tunnel to get more frequent bounces. This will increase your speed. This is necessary in tunnels with low headroom. You can hit left/right while holding the up button.
Note that if you buy a respawn after 500m and use the bunny, this achievement will still count.