Extremely ambitious politicians: one of them killed Shakespeare's Cleopatra (4)
I believe the answer is:
asps
'one of them killed shakespeare's cleopatra' is the definition.
'extremely ambitious politicians' is the wordplay.
'extremely' says to hollow out the word (remove centre letters) (only the extremes of the word are used).
'politicians' becomes 'ps' (I am not sure about this - if you are sure you should give a lot more credence to this answer).
'ambitious' with its centre taken out is 'as'.
'as'+'ps'='ASPS'
Can you help me to learn more?
(Other definitions for asps that I've seen before include "people" , "Pass (anag)" , "Pass around the snakes" , "native Africans?" , "Poisonous snakes" .)