Once you have obtained the Bend Will shout and learned all three words, you have the capability to force a dragon to land, allowing you to ride it. Once it has landed, walk over to its head and the dragon will essentially ask you if you want to go for a ride. Hit
, and you will mount it on the neck, and it will take off. I don't know what the minimum amount of time you must ride for it to count is, but I essentially did it long enough to circle around to land back where I took off, which was about ten seconds. When you are ready to land, hit
again and the dragon will look for a landing spot. On some other sites, it says the dragon will attack you when you land. This did not happen to me at all.
You should be able to use any dragon in Solstheim or Skyrim (I also have the Wyrmstooth mod but did not check there). Once I had completed the DLC (you don't get the third word until near the end), I fast traveled back to Skyrim, and essentially fast traveled between dragon locations until I had my five rides. As soon as you physically mount the fifth dragon, the achievement should unlock. Shouldn't take more than ten minutes total.