To unlock High Speed Pursuit you must complete fifteen pursuits within the game.
Pursuits are a mini collection of goals and challenges. They usually consist of 4-5 goals each. To complete a pursuit, all goals within that pursuit must be completed. You will see a tick next to every goal you have completed. When you complete all goals within a pursuit, every goal will be greyed out and the pursuit will be registered as completed.
You can see how many pursuits you have completed. At the top, it will say Season One (X/107), where X is the number you have currently completed.
There are 107 pursuits within the game currently, some taking minutes, some taking thousands of hours. Luckily for this achievement, you just need any fifteen.
To view your pursuits, press START. On the character screen, hold down LT, and select GOALS (at the 1 o Clock position). Here you can see all pursuits divided into categories.
For those wishing to knock these out fast, below is a list of easy pursuits that can all be completed in ACT1 with little/no trouble.
Browse the list, and see what you find easiest. For some of the combat ones, these require respecing your EGO points.
Mount Tam Exploration 1
Mount Tam Exploration 2
Mount Tam Exploration 3
Mount Tam Exploration 4
Mount Tam Exploration 5
The Truth Behind Tranquility
Mount Tam Recon
Ego Evolution 1
Ego Evolution 2
Snipe from the Shadows
Boomstick to the Face
Living to Kill
Pedal to the Metal 1