Ludicrously busy chap in charge (5)
I believe the answer is:
manic
'ludicrously busy' is the definition.
(I've seen this before)
'chap in charge' is the wordplay.
'chap' becomes 'man' (I've seen this before**).
'in charge' becomes 'ic'.
'man'+'ic'='MANIC'
(Other definitions for manic that I've seen before include "Ludicrously busy" , "Obsessive" , "Frenzied" , "Wildly excited" , "Wildly energetic" .)