If you just don't have a great handle on the keyboard yet, this is the alternative method I used:
Unlock the ability to play keyboard sections on guitar by completing the "Guitar Immortal Challenge". Once this is done, "Keys" will now be a play option from guitar.
Using the guitar, select a song you believe you can complete a 350 note streak on. I played through the Keys section of "Bohemian Rhapsody" on medium and got 100%. I found the song easy due to its steady tempo and my familiarity with it.
Once you've completed your song, return to the main menu and switch over to the Keyboard controller. Then play through a keyboard section of any song. Once you complete the song, the achievement will unlock, regardless of how well you did.
Notes: You will not be able to play the "350 Note Streak" Goal through the Keyboards Goals menu. It gives you a "wrong instrument" message.
The achievement will unlock even if you turn off your console after you 350 note streak (using guitar) and then play a keyboard song later.