In the last manga, Goku is able to transform in Super Saiyan God Red on his own, without the other 5 Saiyans surrounding him in a ritual. My question is, in this transformation, can he use the Kaio-ken on top of it, just as he used it with the Super Saiyan God Super Saiyan transformation (Super Saiyan Blue)?
Your question is a tricky one, my friend. It hasn't been covered in the anime or the manga as to whether or not that is possible, but theoretically it should NOT be.
Why? Because in the anime which has already been declared canon ("somewhat..."), it has been stated that Kaioken on top of SSB is allowable because it is used on god-ki under perfect ki control.
SSR does not have perfect ki control which is why it can only be done via ritual, and also why Vegeta does not have SSR form.
Arguably since the manga and the anime deviate whereas spoilers below for anyone who isn't caught up...
In the battle between Goku and Hit, two different outcomes are portrayed. In the anime, Goku uses Kaioken x10 while in SSB to enable him to overpower Hit's time-skip ability. Doing so makes him unable to control his ki for a period of time, forcing him to warp into the wrong places and leaving him unable to even fly correctly. Now in the manga, Goku transitions between SSR and SSB, throwing Hit's timing out of whack and eventually defeating him. So this brings in two hot topics:
- Is SSR still usable without the ritual due to the zenkai boost?
- Is Kaioken usable compounded with SSB? Which one is the canon here? One is true and the other has to be false, unless for some strange reason both are true.
Kaioken on SSB may not even "technically" be real due to the anime only being partially canon.
TLDR: Theoretically no, and with what has currently been seen in the anime and manga as of June 27, 2016, no. Is it possible? Yes.