To unlock the Speedy McSpeederton trophy in Borderlands: Game of the Year Edition, you will need to complete a lap of the Ludicrous Speedway in less than 31 seconds.
The race track is found in the Dahl Headlands. Unfortunately, there's no visible timer when you start the lap, but there are some shortcuts that you can take advantage of. The video below shows a good route that you can take (credit to SniperWolfHI on YouTube for the video):