To unlock the achievement, you must physically possess $150K. Money earned but not retained does not count towards the achievement. So, money spent before the achievement is earned does not count towards the achievement. This differs from the Baby You're a Rich Man Achievement where all money earned counts towards the achievement regardless of whether it has been spent or is still on hand.
In the below video, Lincoln has just under $149K in the bank. He picks up over $1,000 during a mission so that his total cash is over $150K and the achievement pops: