Caesar's killer you and I dismissed with partner, harshly (8)
I believe the answer is:
brutally
'harshly' is the definition.
'caesar's killer you and i dismissed with partner' is the wordplay.
'caesar's killer' becomes 'brutus' (conspirator against Julius Caesar).
'you and i' becomes 'us'.
'dismissed' indicates named letters should be taken away.
'with' means one lot of letters go next to another.
'partner' becomes 'ally' (I've seen this in other clues).
'brutus' with 'us' taken out is 'brut'.
'brut'+'ally'='BRUTALLY'
(Other definitions for brutally that I've seen before include "Cruelly" , "without any feeling" , "In a savagely violent manner" , "Savagely, violently" , "In a senselessly cruel way" .)