They boast they’ve at last achieved firsts (8)
I believe the answer is:
egotists
'they boast' is the definition.
(boastful or arrogant people)
'they've at last achieved firsts' is the wordplay.
'at last' indicates one should take the final letters.
'achieved' becomes 'got' (get can mean to achieve).
'firsts' becomes 'ists' (Ist resembles 1st).
The last letter of 'theyve' is 'e'.
'e'+'got'+'ists'='EGOTISTS'
(Other definitions for egotists that I've seen before include "Conceited and self-centered people" , "People with exaggerated notions of their own importance" , "They have an exaggerated sense of their own importance" , "Self-centred self-seekers" , "The selfish" .)