Partial ROM or Trap Bar deadlift with strengthen your legs. Really, any partial range of motion movement that you can get strong with will help you pull faster. I know people who can squat a lot more than me who struggle to pull a 600lb chain, that I can basically jog with, for example.
For the grip youre going to have to train with a thick bar for your deadlifts.
For unilateral work you can carry rocks or weights up a hill. Lunges wont help you (do you lunge with the truck, no). If you can do farmers carry up a ramp that will get you strong and work your grip.
But it might be a good idea to practice with a lighter load for longer distances.