He was bound to begin but not to end his act (7)
I believe the answer is:
houdini
'he was bound' is the definition.
(Harry Houdini was frequently tied up or bound)
'begin but not to end his act' is the wordplay.
I cannot really understand how this works, but
'not' could be 'o' (I've seen this in other clues) and 'o' is found in the answer.
'act' could be 'din' (din is a kind of act) and 'din' is present in the answer.
The remaining letters 'hui' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
'to' acts as a link.
Can you help me to learn more?
(Other definitions for houdini that I've seen before include "tie-break ace" , "Harry -, US escapologist" , "Harry was famous as an escape artist" , "one who'd try to escape?" , "Escape artist, d. 1926" .)