Is it possible to get the Rinnegan if you are not in the Senju or Uchiha clan (including branch clans such as Uzumaki) and inject 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.
My answer is yes, but only by nature of technicality. This method is possible, though highly improbable. It may in fact not have a prayer of working at all. However, let me lay out this plan for you.
Firstly, we know that creatures like the Tailed Beasts can be sealed within a host body. The host body then can gain the power of the sealed Tailed Beast. We also know that a human soul can be sealed, though whether it is possible to seal another person inside one's self is not explicitly shown (to my knowledge anyway). We have, however, seen the souls of all the first four Hokage being sealed within the Shinigami as part of the Reaper Death Seal. Putting these elements together, I think it would be safe to assume it is at least possible for one person to use a similar sealing method as used on the Tailed Beasts to incorporate another human being into themselves.
We also know from Obito and Kakashi that the entire chakra and essence of a person's Kekkei Genkai can be placed into another person's body. This is what allowed Kakashi to make use of the full power of Obito's eyes as well as Susanoo.
Now, for the final piece of information before this can all come together, we know that the requirements for activating the Rinnegan, which, as explained by Black Zetsu, is to bring together the chakra of Indra and Asura, thus giving you the chakra of their father. Therefore, if you could actually seal whoever the current successor of Indra and Asura are, then in this given hypothetical scenario, a third party could in fact gain a Rinnegan.
In no shape or form would I consider this natural, and I'm sure there are a lot of questionable morals involved with the entire idea, but through this strange course of action or something similar in nature, I believe it is possible to gain the Rinnegan even as someone who was not born with either Indra or Asura's chakra.
I know, it's a crazy, loophole kind of a way of going about it. But hey, you asked for possibility, not plausibility or even likelihood. Heh.
No it’s not possible, Orochimaru probably tried countless of times to replicate the scenario, but only Madara got lucky; Madara, while being alive, was able to unlock the Rinnegan near his death.. He was reincarnation of Indra was well...and probably had the purest connection to Indra next to Sasuke.
We know in order to get Rinnegan, the person must have Indra and Ashura cells (not just Uchiha and Senju, because the person needs to be more related to Hagoromo), But it’s extremely hard to match the right genes. Madara, being the reincarnation of Indra and Hashirama being reincarnation Ashura is your best chance...other Uchihas will not even come close. After that, it will take years to even unlock it, and who know what events need to take place... I think Obito had Hashirama cells in him if I recall, but he was never able to unlock Rinnegan.