There are different interpretations of Candrakirti, even among Madhyamika-Prasangika scholars. That said, Prasangika scholars generally hold that Buddhapālita (470 AD - 550 AD) gave the first Prasangika presentation in the form of a commentary to Nagarjuna and Aryadeva, but is not the founder of the Prasangika school. Bhāviveka (500 AD - 578 AD) then refuted Buddhapālita and is to be regarded as the founder of the Madhyamika-Svatantrika school. It is thought that they did not debate face to face because Buddhapālita belonged to a lower class. Candrakirti (600 AD - 650 AD) then established the Madhyamika-Prasangika school by way of refuting Bhāviveka so-called Svatantrika positions. Candrakirti is held to be the founder of the Prasangika school (over Buddhapālita) because he established it through properly refuting another. It is sometimes thought that Candrakirti was Buddhapālita's reincarnation. It is also sometimes thought that he debated Bhāviveka face to face (although their commonly established date of their birth and death would not allow it)
As I said, there are different readings of Candrakirti, even among Madhyamika-Prasangika scholars. These differences are significant, and are in relation to:
- The notion of other-emptiness & self-emptiness;
- The nature of emptiness (whether it is an affirming or a non-affirming negation)
- The way emptiness is realized (implicitly or explicitly)
- The meaning of 'prasanga' and that of 'svatantra', and so forth.
I suggest that, first of all, you study the context and the possible meanings of the so-called Svatantrika / Prasangika distinction. Sara McClintock & Georges Dreyfus's The Svatantrika-Prasangika Distinction: What Difference Does a Difference Make? is interesting.
The two main Prasangika positions are:
Gorampa is known for criticizing Tsongkhapa's positions: he goes as far as to say that Tsongkhapa is in fact a proponent of Svatantrika tenets. Thus, studying Tsongkhapa in order to know what Gorampa refutes would be most relevant.
In relation to studying Tsongkhapa, I advise you to read Tsongkhapa along with his main disciples: Gyaltsab Je and Khedrup Je. Khedrup Je is usually easier than Gyaltsab Je. Tsongkhapa's commentary to Candrakirti was translated by Geshe Kelsang Gyatso, I will thus not expound on that. Also, from a Geluk perspective, we do not usually study Tsongkhapa's (or any Prasangika commentery to Candrakirti). We study Svatantrika commentaries to Candrakirti's Madhyamakavatara (i.e. Haribhadra, and so forth). Our gateway to the Prasangika system is other. It is Tsongkhapa's commentary to Nagarjuna (the Ocean of Reasoning), it is the Special Insight chapter of the Lam Rim Chen Mo, along with Tsongkhapa's Final Exposition of Wisdom, Gyaltsab Je's commentary to Shantideva (Shantideva also presented as a Prasangika himself). Having studied them, having understood them thoroughly, you could go a step further and read Candrakirti's root text (Madhyamakavatara) for yourself, and also read Gorampa's positions and his followers.
In relation to studying Gorompa, you may read
Ju Mipham, whose presentations accord with Gorampa's. Ju Mipham's commentary to Chandrakirti's Madhyamakavatara is most relevant.