Account by first person achieves the climax (4)
I believe the answer is:
acme
'the climax' is the definition.
(I've seen this in another clue)
'account by first person' is the wordplay.
'account' becomes 'ac' (abbreviation).
'by' says to put letters next to each other.
'first person' becomes 'me' (first-person pronoun).
'ac'+'me'='ACME'
'achieves' acts as a link.
I am not very happy about this link. It may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for acme that I've seen before include "crisis in history" , "Summit of achievement" , "Summit, crowning point" , "Point of maximum development" , "Pinnacle" .)