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.
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).
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
Clare being 1/4 Yoma surely indicates that she has full potential to house 3/4 of human energy mind being amongst the 1/4 Yoma.
Most Claymores, in a sense, have it rough, you might say, as they are all equally assessed in the middle of battleground between lethal killer and calm human, exceptions such as Teresa and Priscilla (before becoming aggravated by Teresa) are able to house full 100% human control within their means of reach of fighting, until they begin to access Yoki.
Clare is an exception to being discussed over ranking! This Claymore Universe may indeed need to stop entirely placing Clare on a parallel comparison to the other Claymores, as this is why:
She became a Claymore to exact her spiritual bond connection to the physical beings of herself and Teresa's physical being, therefore she has no hatred whatsoever towards Yoma or Claymores. Only Priscilla, not Priscilla the Awakened Being, just Priscilla as an Entity, which qualifies Clare to have a unique case within the organization, as all other Claymore Creation stories, involve traumatic events with Yoma or Yoma Awakened Beings that cause them to choose to exact revenge (offensive) or protect themselves (defense), from the Yoma beings.
The fact she took in 1/2 already Yoma Potent Flesh versus 1 whole piece of Yoma flesh, deems the obvious we know, 1/4 Yoma in Clare. This alone should not even begin to have a place in conversation over ranking, as with her fact she had 1/4 Yoma, means she has more Human Energy Control, which finalizes the logic behind her consistent ability to awaken far many more times than her comrades did, and with seeming ease reverts back more frequently, which wouldn't be possible with any half bred as by that many awakenings they would have by then surely lost their 50% control.
Clare needs to stop being judged. She's practically Teresa's daughter, that in itself gives her enough credit to be called a Badass Warrior.
Claymore are ranked based on their combat ability. Their ability is taken from a number of factors such as strength, Yoki, and intelligence to name a few. Claymores typically only change rank by challenging and defeating a higher ranked Claymore or when a Claymore dies or awakens.
Now, just because a Claymore dies, it doesn't mean others will go up in rank. It is entirely possible that the newly created Claymore takes the exact rank of the one that died or awakened. The reason that Claire never advances in rank is because she is only one quarter Youma since she was created with Teresa's remains and it is said that she'll never achieve the potential of a regular Claymore that is one half Youma.
I think comparing Clare's baseline overall power, speed and combat ability, she's no match to either Teresa or Priscilla, or even the top 10 or say top 20 of Claymores.
But in a half-awakened state, it is possible that she can beat Priscilla and all the other strongest Awakened Beings, nothing more to say about all the Claymores from rank 2 to 10. Her powers are specifically developed to fight Awakened Beings. And her greatest strength is her ability to develop a part of her body or skills to match the power of the Awakened Being she fights with, e.g. Rigaldo.
As with Teresa, I think she is the strongest of them all, that is a special status relegated to her. We'll never know the extent of her powers when she became awakened. I think that Clare, if she fights with Teresa in her awakened state, would be able to find a way.