You're given a Bone Gryphon which comes with four abilities (listed in the order they are located on the action bar):
1) Frost Breath - instant primary spam attack; no cooldown
2) Plague Strike - deals a small amount of direct damage and applies a DoT; has a 12 second cooldown
3) Bone Armor - instantly gives your gryphon four "bones" that absorb damage (basically a shield); has a 20 second cooldown
4) Speed Burst - doubles your flight speed for 8 seconds; has a 1 minute cooldown
Head over to Onslaught Harbor to the west of Death's Rise and find a safe spot to land, then use the the Bone Gryphon to hop on. Use Bone Armor whenever the cooldown is finished. When fighting the Onslaught Gryphon Riders, start off with Plague Strike and spam Frost Breath until the rider is dead. Kill ten riders to complete the quest.
If your Bone Gryphon's health is low, find a safe spot to dismount and re-summon it. You'll hop on a new Bone Gryphon will full health.
When turning in the quest, make sure to dismount your Bone Gryphon and use your own mount to fly back to Death's Rise. If you leave Onslaught Harbor, you will get a warning that says you might lose your Bone Gryphon. Although it's possible to get to Death's Rise while on the Bone Gryphon, you should just play it safe and use your own mount.
Of course, you can always kill the riders the old fashioned way: with your own weapons and spells. :)