Listen to scat singers and pay attention to the sounds they make and the techniques they use.
Learn the arpeggios of chords, which are the notes that make up a chord. For example, the notes that make up a C major chord are C, E and G. Scat singing relies on arpeggios, so you'll need a strong knowledge of arpeggios for success.
Sing scales and arpeggios, using sounds like "de" and "doo" and "dut." Sing these notes in a staccato manner --- short and detached from other notes -- and try different sounds. Remember, you're trying to imitate an instrument's sound.
Sing different combinations of notes and sounds. You don't have to just sing chromatically. Try out different note combinations to see what you think sounds good.
Sing along with a jazz or blues recording, adding a scat solo at an appropriate part. Continue to practice by trying out different sounds and notes.