Your celebrity rank is determined by the number of fans you have.  In order to climb the chart, you must obtain MORE fans than the celebrity ahead of you. Even if you complete several gigs with 5 stars in a day, if the overall number of fans does not surpass the character above you, you will not advance up the chart. 
For example, if you are sitting at the #150 spot with 100k fans, and the celebrity above you at #149 has 200k fans, you will not move up until you have obtained at least 200.1k fans. To view this chart, simply tap on your Top Stars icon in the upper right corner of the screen and scroll down until you see the character above you.  From this list, you can view how many fans your competition currently has. You must calculate from this guideline so you know how many fans you must acquire in order to move up the chart.
Be sure to play daily to help increase your fan numbers!