You can grind this achievement but will take lots of time and dedication.
There is a fast way to achieve this.
First of all, find one of this plants:
http://imgur.com/a/rVfDGYou can find them in the snowy floating islands above and around teleporter 7.
When you succesfully found one, grab the big leaf with both hands and start trying to bend it down. You want to use the plant like a catapult. At some point of the bending process, the plant will launch you in the air really far away and if you remain attached to the leaf, the leaf itself will be launched with you and this will count as climbing because you are still attached in climbing mode with your hands. You will probably need 2-3 plants to finally reach the distance of 10000m depending on how far you fly away before losing the leaf.
P.S: Your character can have some strange visual glitch during the launch, and sometimes the leaf will not be lauched with you. Just try again with another plant.