Obtaining any kilometer achievement is much easier than it seems, you will do a ton of running...... A TON of running in this game. Bad news is don't trust your achievement tracker on the ONE as it is absolutely worthless, also the achievements aren't even popping for me, they just unlock without saying it so if you worry about it not counting don't worry you are fine.
You can view your current amount of Kilometers ran in the menu where you are viewing your character on the right. If you plan on enjoying this game don't go out of your way for any of these achievements until you are done with everything else.
There is a Mini-bike in the game that should count for this achievement if they coded it correctly because it says Travel 1000 Kilometers without saying a required method. The Mini-bike is known to be extremely glitchy in Co-op/online so recommend doing this offline with Zombies turned off.
To make a Mini-bike you must find the Schematic which is random, higher chance of finding it in a crate or Safe. Once you get the schematic, save up Small pipes, duck tape, pelts (kill bears and deer). You will also have to make a Forge once you get the seat made and Handle's made. The Mini-bike will make this go by faster, once you do get it find a town/city and drive it in circles to accumulate KM.
You will also need tons of fuel so save up as much as possible, you can also breakdown Drum's for 600 gasoline so make sure to grab those.