There are two steps to this achievement.
Step One: Obtain Brainiac
*** Spoiler - click to reveal ***Once you are able to use Remote Links In Gotham Free Roam (complete level four in story mode), you need to use the one in Gotham Park (Map Icon shaped like a tree). You can do this as soon as Remote Links are available in Free Roam Story or wait til after you unlock Free Roam Free Play (unlocked by completing all 15 levels in story mode; after the credits roll). This will reveal the location of Brainiac. Mark him on your map, travel to him, defeat him and purchase him for 500,000 studs.
Step Two: Defeat a Lexbot
*** Spoiler - click to reveal ***If you have already unlocked Free Roam Free Play, switch to Braniac as soon as you purchase him. Now fly to the top of Wayne Tower (the only location guaranteed to spawn Lexbots). As soon as they appear, they will rush you, just beat one down and the achievement will pop!
If you are doing this in Free Roam Story, travel by foot or vehicle to Wayne Tower and go to the rear of the building. On one side is a climbing wall that can be used to reach the top (necessary for the Gorilla Thriller Achievement). Once you reach the top, the Lexbots will spawn and rush you. Defeat a certain number and one will surrender, giving you the option to purchase it. Do so, then use the Remote Link on top of the tower to Fast Travel back to the Bat Cave. Use the computer to enter any previously completed level in Free play. As soon as the level starts, switch your character to the Lexbot by holding Y and selecting him from the character menu. Next, switch to your second character by tapping Y and use the same method to switch him to Braniac. Finally, beat down your Lexbot partner and the achievement will pop! Alternately, you can skip buying the Lexbot, use the Gotham Park Remote Link to return to the Bat Cave and enter a Free Play level you know contains Lexbot AI enemies, switch to Brainiac as above, and once one shows up in the in the level, defeat it and the achievement will pop!