Hit 250 notes in a row, with your streak carrying over from one song to the next, over a minimum of two songs. For example, you could hit the last 200 notes of a song, then pick another song and hit the first 50 notes of that. You can split it however you want though, and you can join together more than two songs.
You can get this in Learn a Song, Nonstop Mode, or Lessons. (Update: possibly Score Attack too, see the comments)
If you want to play it super safe you could level down a few songs to a low difficulty and work through those (to level down: start the song, press B to open the Riff Repeater and move the Difficulty slider all the way down, or as low as you feel comfortable with). Of course, the lower level you play the fewer notes you'll get, and the longer it'll take to hit 250.
Note: it's not your longest streak that gets counted; it's the streak you're running at the start/endpoint of the songs. So if you miss the first note in your second song, you've lost the achievement and have to start again.