For the completion you have 3 challenges,
1. Don't use the winch
2. Use the cockpit camera when driving with the trailer
3. Deliver the Skidder within 6 minutes
1 and 2 are simple enough, don't press Triangle and as soon as you start hit in the Left Stick. Take off and go about your path.
3. Gotta go fast. the problem areas I had were turn 1 in the first section of the dirt, stay on the left side of the road as the right will make the trailer bottom out. When you get the left turn near the house stay to the outside as the inside will make you roll over. 2 turns after that during the S section you can bottom out the trailer on the inside again. I messed up here and got stuck but reversing and trying again to continue works. There on I left the gear in 1+ and the momentum kept the truck moving. I ended up with a time of 5:01.
(Got the achievement on the X1 version of the game.)