At first glance, I thought this was going to be hard, but it only took a bit of practice to gain the muscle memory to play up and down the scale. While a major scale is 7 notes using the "Major" interval. The Chromatic scale is all 12 notes the "white and black on a piano" between one C to the next.
C-D# uses a 4 note pattern on hole 1, I'll call X blow, blow/slide, Draw, Draw/slide
E-F# uses a 3 note pattern on hole 2, we will call Y blow, blow/slide, Draw/slide
G-A# repeats on hole 3, pattern X : blow, blow/slide, Draw, Draw/slide
B-C is played on hole 4 draw and 5 blow we can call pattern Z.
By ending on 5 blow you are set to travel another octave up using pattern X.
Note, a chromatic scale in any key can be played by simply starting and ending on that note and playing these same notes.