Is it possible to get the Rinnegan by injecting Senju and Uchiha DNA into yourself?
To awaken the Rinnegan, it's not enough to be a Senju with Uchiha eyes or vice versa, you must be Indra's successor that obtained chakra from Asura's successor.

A successor does not necessarily come directly from the Senju clan, as Naruto, who is an Uzumaki, is a successor. However, Indra's successor is always an Uchiha.

So in short, no. It's not possible to do what you describe.