Once you've completed the game go to chapters, chapter 5, practice speed run and I would suggest running it on easy.
The platforming is pretty straight forward once you know the route. I came in a 7 minutes and 10 seconds and messed up a lot. Try and get more headshots than I do in the below video.
Knowing where the enemies are is key to the middle combat section. Once you've cleared the first few go for the Wes 44 Magnum behind the washing machines to make the others quicker.
The video was only my second attempt but I reckon with practice 6 minutes is easily achievable.