When starting out as an improviser, the half diminished 7th chord is often a difficult chord to hear and play over melodically. Even after working on it for years the half diminished chord can still be more challenging to sing over or to play by ear over than, say, a Major 7th chord. This simple exercise outlines just the 4 notes of the half diminished 7th chord in both inversions and broken arpeggios. Singing of the material first, to make sure you can really hear what it is you’re working on will insure continued growth over the long haul.
You’ll often find the half diminished chord in a ii-7b5, V7b9, i-7 progression. In this context the ii-7b5 is nearly identical (in terms of function) to IV minor chord (as found in a IV minor, V7, i cadence common in classical music). The main difference is that the 6th degree of the IV minor chord is in the bass, thus creating a half diminished sound.
Once you are comfortable with the the exercise, try improvising for a while in each key. Over time, combined with singing the material to train the ear, this simple exercise should help to clarify the half diminished sound.