Unlock the ability first
Before you can ride a Graug you need to make sure you unlocked this ability. The ability will be automatically unlocked when you play certain missions.
Sneak behind the Graug
The first step is to sneak behind the Graug without letting it notice you.
Injure the Graug in its knee
When you are behind the Graug you will be given an option to injure it in the knee. This weakens the Graug and makes mounting it possible.
Shoot the Graug in the face
Next, you need to stun the Gruag. You need to shoot a charged arrow in its face. On PS4 this can happen by holding both L2 and R2 then releasing R2.
Come close to the Graug again
Come close to the stunned Graug and press the mount button, R1 for PS4. This button will flash on the screen as soon as you come close to a stunned Graug. (See How to interrogate enemies in Shadow of Mordor?)
Follow the sequence of buttons
As soon as you mount the Graug, a sequence of buttons will show up on the screen. Press that sequence in a timely manner. Missing one button can get you killed.
Do the final move
The final move is always that red round circle with the arrow inside. Here is exactly how to do it.
Optional: Use shadow strike
In case you unlocked the ability of instantly mounting animals using the shadow strike, you can use it to bypass many of the steps described above. (See How to do a strike from above in Shadow of Mordor?)
How to ride a Graug in Shadow of mordor?
How do you mount a Graug in shadow of Mordor? Riding the Graug in Middle-earth: Shadow of Mordor