Each bot has the ability to rank up to level 10 by earning XP, which is obtained for each bot that you have active for a game (up to three at a time). The amount of XP you earn is 10% of your total ending score. You need to use each robot enough to earn the sufficient amount of score to level each up to 10. I am not sure of the exact amount, but you will need to earn a cumulative point total far into the millions for each bot. You will have to play many, many games with a given bot equipped. It is recommended that you don't use a bot anymore once reaching level 10 for it. Keep switching bots to maintain progress towards this achievement.
It may be worth it to invest in power crystals, which can be purchased in a shop. They are single-use items that multiply all of your XP in a game by 3. The crystals are pricey, but if you use one in a game where you earn crazy score, they pay off. The crystals only last for one game, and are applied in the bot selection screen before starting the game.
NOTE: Bots used have a charge that drains with use, and must be replenished before using again. The charge slowly replenishes with time, but can also be instantly replenished using batteries purchased with your hard-earned coins (the latter is not recommended, considering how slowly coins are earned and how many you need for "Catch Them All").