During the story of this DLC, you will learn a new shout called 'Bend will'. You must then use this shout at its full power(holding down ) on any dragon that you may encounter. The dragon will then land allowing you to mount it by walking up to it and activating it with . When you have mounted the dragon, it will proceed to take off and fly in its own fashion, as you cannot control its flight path. To dismount the dragon, press at any time and the dragon will find a suitable place to land for you to dismount. Repeat this 4 more times and the achievement is yours.
You first get the oppurtunity to do this in the final story mission 'At the Summit of Apocrypha' and the dragon you use does count towards the achievement. A good way to do this is to fast travel to any dragon shrine locations you may have discovered and use your 'Bend Will' shout on any dragon that may be present.
This can also be obtained whilst in the original Skyrim world, as well as in Solstheim.