Rhino Van Dam has a good solution here, but I just unlocked this achievement and I'd like to add some advice.
First, you can use a couple of gadgets to make your life easier. I used the Turbo Boost and Dezapinator gadgets - the final two in the gadget stash. Turbo Boost causes boost rings to appear every now and then while you're playing. When you hit a boost ring, all the obstacles for a couple hundred meters are destroyed. This can save you a lot of trouble towards the end of 5000m. The Dezapinator causes the yellow zapper links to randomly fail about 10% of the time. This can also save you in case you can't maneuver out of the way in time.
Second, try and make sure to hit every single vehicle token you come across. They effectively reset the difficulty for a few hundred meters and give you an extra hit before you die.
I don't recommend using the Head Start items. When I used them I often found the change of pace when they wear off is jarring and caused me to die very quickly. I found it much easier to start at 0 and take my time.
Last advice echoes what Rhino said. Try and catch spin tokens, if possible. With luck you'll spin a revive or blast. Bring a Revive and Final Blast and use them only if you get pretty close to 4000m or so.
Good luck - this one is pretty tough!