Simply find all upgrade chips for one gun, and purchase the upgrades. There are other achievements for finding all chips; please see those to find them.
You will need to upgrade all weapons in order to complete the game anyway, so this achievement is pretty much unavoidable. Only tip I have is to not rush into spending money on the first gun ASAP, since it's the worst one. Upgrade that one last and focus on the second gun or your health (or whatever other chips you find) instead.