To start with I would suggest entering the commands to enable certain traits, this can be done by pressing RT,LT,RB & LB at the same time then entering in the following:
These don't disable achievements, so I would highly recommend them.
The Entertainer career requires either a musical instrument skill or comedy skill. Now you could power train them before you start work or just train up in between your work shifts, entirely up to you.
For each promotion their will be a skill level requirement and also a daily task to improve your job performance. To start with these daily task will just be to play an instrument or write jokes. Once you have gained a few promotions though there will be a choice to become a musician or comedian.
Both paths are straight forward requiring the same daily tasks as before but obviously restricting the options. I personally chose musician as my sim tended to just pick up an instrument without me even needed to direct them.
Optionally I found having sims energized and focused helped with job promotions, so drink some coffee and play a bit of chess before going to work if you wish.
Once you have gained the final promotion you will gain the achievement.
As a side note, in regards to any Sims 4 achievements not unlocking, I have found a solution that I had to use a few times. To unlock anything follow the steps below:
Remove profile from Xbox.
Reset Xbox via settings.
Load up the save in which you have earned the achievement but it has not unlocked.
It should pop before it finishes loading.