*Warning* Attempts to do this on a Mac using EndNote X3 were not successful.
To add in text citations and a formatted bibliography in Google Docs using EndNote, first drag and drop the formatted citation from EndNote. When you are done with your document, download the document as a word document and format the paper as you normally would using EndNote's Cite While you Write, or download an RTF file and format the paper using EndNote. Step by step directions for downloading RTF are outlined below. EndNote also has a detailed training video for the RTF method here.
To add a formatted citation, highlight the reference in your EndNote library, then drag and drop or copy and paste on to Google Docs.
Once your document is complete, download as an RTF file to format your citations and bibliography. In EndNote click on Tools -> Format Paper -> Format Paper.
Select the document and in the pop-up window, verify the citations, select your citation style, and click Format.
Save the document. EndNote defaults the document name to have the name plus the citation style added to the end. Upload the document back in to Google Docs. Be sure "Convert upload files to Google Docs format" is turned on in your settings.