We have a very brief introduction How does Frege's definition of number solve the Julius Caesar problem? here on SE. Zalta's review of it for SEP is a good place to start more scholarly research. Dummett's book Frege: Philosophy of Mathematics is a comprehensive classical commentary. In 2005 Dialectica devoted a whole issue to the Julius Caesar problem, with Heck, Kemp, McBride and Tappenden contributing. Tappenden's entry gives a good historical background. I hesitate to recommend the order of reading since it depends on the specific scope and goals of the project, hopefully descriptions below will help with selecting from the many references.
Aside from Heck and Hales-Wright, who defend the neo-Fregean solution, Zalta cites his own Natural Numbers and Natural Cardinals as Abstract Objects and Wehmeier's Consistent Fragments of "Grundgesetze" and the Existence of Non-Logical Objects (1999):
"Until recently, it was thought that Frege solved this problem in §10 by restricting the universal quantifier ∀x of his Gg system so that it ranges only over extensions... However, recent work by Wehmeier (1999) suggests that, in §10, Frege was not attempting to restrict the quantifiers of his system to extensions (nor, more generally, to courses-of-values). The extensive footnote to §10 indicates that Frege considered, but did not hold much hope of, identifying every object in the domain with the extension consisting of just that object."
Greimann's What is Frege's Julius Caesar Problem? (2003) places the problem into the broader context of Frege's other concerns, and Kim's Strengthening of the Caesar Problem (2011) for a critique of the neo-Fregean solution:
"The neo-Fregeans have argued that definition by abstraction allows us to introduce abstract concepts such as direction and number in terms of equivalence relations such as parallelism between lines and one-one correspondence between concepts. This paper argues that definition by abstraction suffers from the fact that an equivalence relation may not be sufficient to determine a unique concept. Frege’s original verdict against definition by abstraction is thus reinstated."
Salmón's very recent Julius Caesar and the numbers is an exegesis of how Frege's original intentions could be reinterpreted:
"Dummett and other interpreters have seen in Frege’s criticism a demanding requirement on such definitions, often put by saying that such definitions must provide a criterion of identity of a certain kind (for numbers or for linear directions). These interpretations are criticized and an alternative interpretation is defended. The Caesar problem is that the proposed definitions fail to well-define ‘number’ and ‘direction’... A minor modification of the criticized definitions well-defines ‘0’, ‘successor’ and ‘number’, thereby avoiding the Caesar problem as Frege understands it, but without providing any criterion of number identity in the usual sense."
Even more recent Rota on Mathematical Identity: Crossing Roads with Husserl and Frege by Christopoulou (2019) gives an interesting comparative analysis with the phenomenological approach of Husserl and Rota.
"In this paper I address G. C. Rota’s account of mathematical identity and I attempt to relate it with aspects of Frege as well as Husserl’s views on the issue... I detect certain similarities among Rota’s views and Frege’s account of the constitution of arithmetical identity on the grounds of 1–1 correspondence."