Artist to draw relentlessly competitive way of life (3,4)
I believe the answer is:
rat race
'competitive way of life' is the definition.
(I've seen this before)
'artist to draw relentlessly' is the wordplay.
'artist' becomes 'RA' (Royal Academician, member of Royal Academy of Arts).
'to' means one lot of letters go next to another.
'draw relentlessly' becomes 'trace' (synonyms. I am not sure about the 'relentlessly' bit.).
'ra'+'trace'='RAT-RACE'
Can you help me to learn more?
(Other definitions for rat race that I've seen before include "Competitive struggle for wealth or power" , "Continual round of hectic competition" , "hurly-burly of competitive work" , "Grim struggle" , "struggle to get on in the world" .)