ChakraCore will only feature the core set of capabilities of Chakra itself, and won't offer Chakra's private bindings. From Microsoft:
ChakraCore shares the same set of capabilities that are supported by Chakra in Microsoft Edge, with two key differences. First, it does not expose Chakra's private bindings to the browser or the Universal Windows Platform, both of which constrain it to a very specific use case. Second, instead of exposing the COM based diagnostic APIs that are currently available in Chakra, ChakraCore will support a new set of modern diagnostic APIs, which will be platform agnostic and could be standardized or made interoperable across different implementations in the long run.
Microsoft says that they will open a public GitHub repository for ChakraCore in January 2016. They plan to provide more details and guidance on how developers can best contribute to the project. Developers will be able to contribute everything from new features to bug fixes to the project, and give suggestions for improving ChakraCore over time.