For this achievement, first you need to find your way to an End City. To do this you must defeat the Enderdragon, which will open up a portal around the edge of the small island leading to the outer islands of The End which are 1000 blocks away.
Once you reach the End City, you will find Shulkers attached to both the inside and outside of the towers. The Shulkers will shoot a projectile at you which will cause you to float upwards for a few seconds. I found it easier to go up the outside of the tower, as that meant i didn't get caught on anything, and then landing on the roof once I'd gone high enough. Apparently you can also throw a spare Ender pearl high up the building once you start floating although it was easy enough that i didn't need to do that/didn't try it.
Finding the End City's can be difficult and you can find yourself wandering for hours without seeing any at all, therefore i have put the seed i used below with the locations of some End Cities that i found. The End portal is quite far away from the spawn however, but if you are stuck, or like to plan ahead, you could try using this.
Seed: 173100852
End Portal Location: 4605, 36, 2118
End City -1900 , x, -200
End City -1484, x, 650
End City -2450, x, 332
End City 417, x, 1068
End City 670, x, 1300
End City 1400, x, 723