Customize upload on REST API
If I want to customize how files are uploaded when a REST call is received, what package/class should I be looking to modify? What's a good starting point?
Also, once the code is modified, how do I recompile it and deploy it?
Using the Nuxeo Platform, you never have to (and should never) modify Nuxeo source code: everything is customizable by extending or overriding Nuxeo. You should first start reading Customization and Development documentation. That's the extensibility point where Nuxeo make it easier to understand, upgrade, customize without breaking, … and so on.
About file upload on REST API, your question is too much generic: do you want to customize default behaviors of current services, implement your own logic, add your own REST API, …? You may want to give a look at Simple REST, Content Automation, Import, Bulk document importer, WebEngine (JAX-RS), or the Nuxeo Platform Explorer …
Once again, you don't need to (and should not) modify the source code but if you still want to do it, then read Working on Nuxeo sources.