How does a Claymore move up in rank within the Organization? How do Claymores advance in ranks within the Organization? 

We see many kills during the whole TV show, but Claire never advances in rank.
It is worth bearing in mind that the TV show only covers the first 7/8 books, and it might be addressed later on in the story.

As far as I could find however, the ranking system seems to be done by generation rather than reallocating ranks when someone dies.
You can see claymore occasionally contest someone's position in the manga, but it doesn't seem to be an official thing.

Ranks by Generation


  Claymore warriors No. 1 through 47 are ranked on their baseline Yoki potential, strength, agility, intelligence, sensing and leadership. A warrior's rank rises and falls according to the warrior's strength in relation to other warriors.
  Claymores are known to belong in different eras/generations, marking the time they joined the Organization, but not all Claymores who are active at the same time belong to the same generation. This is because Claymores never age beyond their physical prime.


It's unclear as to when a generation changes, as some claymore survive more than one.

We can see on the linked site that one generation's order has changed.

The old generation (旧世代) (Teresa era テレサ時代).  


No. 1        - Teresa  
No. 1 and 2  - Rosemary  
No. 2  - Priscilla 
No. 2 and 3 - Ilena 
No. 3 and 4 - Sophia    
No. 4 and 5 - Noel 
No. 5 and 6 - Elda


Recalling Priscilla who was a newly created claymore who shot up through the ranks to become number 2, I would imagine that in exceptional cases such as these, the organization would alter the ranks, but mostly it would be left as is until the next generation comes around and new claymores are assessed