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 Writer career requires 2 skills, writing and either charisma or logic. 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 read books and then progressing to writing books. Once you have gained a few promotions though there will be a choice to become a author or journalist.
The journalist daily tasks are to write articles, which requires you click on another sim selecting the interview option and then write an article about them. This can be annoying if you have a small household, where other sims might be at work etc. I would not recommend this option.
The author daily tasks are a lot simpler and less time consuming, these are to write books by using a computer, as previously. Much easier =)
Author requires logic, so bear that in mind.
If you do want to go down the journalist route, the skill requirement is charisma.
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.