Challenge mode can be found under Single Player - Arcade.
It involves performing kills in a certain time and style for each checkpoint.
An example would be (20 seconds - Explosion Kills only - No Pills).
In order to earn a 4x multiplier you need to complete a challenge at each checkpoint.
Your multiplier will increase with each successful attempt but if you fail, it will reset and you will need to restart. The first chapter is the easiest to achieve this one.
Just complete the chapter as normal, using bullet-time to your advantage and complete the necessary challenges required. You can also use Soft-lock if you prefer as Free-Aim is not a requirement. It will unlock once you complete the challenge at the parking lot.