Before you can race Razor you first need to complete 9 race events, 5 milestones, and have a total of at least 10,000,000 bounty. Here is the easiest way to complete these requirements (Note: I used the Lancer Evo for these and for the final 5 races against Razor):
Races: Do all of the sprint, speed trap, and tollbooth races. The circuit races aren't too bad either. Basically if a race is too much for you just try another one (though sprint and speed trap were always the easiest for me.
1) Do the speed camera/trap ones first (they are the easiest). If you cannot hit the right speed at the camera, restart the milestone and back up a fair bit to give more distance to reach the necessary speed. Also, hit the NOS boost as you approach the camera. There are three of these.
2) Complete the pursuit lasting under 2 minutes milestone. Easy way to do this is to go to the bounty section of the Blacklist 15 menu. Choose the location nearest to the baseball stadium (should be the one in the 2nd row on the left). Once you start, immediately head for the baseball field and hit the pursuit breaker. Hide in the stadium. This can easily be done in under 2 minutes.
3) Complete the pursuit lasting over 13 minutes milestone. The easiest way to do this is the take advantage of the very popular bus stop glitch (you can find videos of this on youtube). Make sure you do this in a car that doesn't have a lot of heat already on it (ex. a car you just won from the blacklist or purchase a new one). Go to the bounty section of the Blacklist 15 menu and select a location relatively close to the bus station. Get up on the platform and wait for the 13 minutes to unlock the milestones. If you find the cops abandoning the chase, drop down, re-engage the cops, and return to the platform. Back up should be called in and you should just be able to sit on the platform for the rest of the 13 minutes. Then return inside the bus station and park in the middle of the catwalk with the class barrier. Wait for cooldown to finish!
I already had 10,000,000 bounty before starting the requirements to race Razor, but if you need more just use the bus-stop glitch.
Racing Razor is fairly straightforward, and the race likely to give ppl trouble is the circuit race (2nd last race). After a few attempts (and learning to use short cuts) you should win this. After finishing all 5 races with Razor the achievement should unlock during the cutscene
***EDIT 12/09/2011***: As someone suggested, here is a link to a video showing the bus stop glitch: