There are numerous posts that have explained residual deviance and parameter estimates for the quasi Poisson. But since there is no probability distribution pertaining to the quasi Poisson and hence no likelihood, how exactly is the residual deviance computed?
Here: https://cran.r-project.org/web/packages/bbmle/vignettes/quasi.pdf is a good discussion of this. Fitting a quasi-poisson model means to fit a model with a poisson likelihood function when you do not believe that the data are generated from a poisson model. When do you really, really believe in such things? Anyway, in most cases, models are at best good approximations. To use a quasi-poisson model, you must at least believe that some aspects of the model makes sense. So, we could say that most uses of likelihood models are in reality only quasi-likelihood (the most important clear exception when you analyze data simulated on the computer by a known model. Known, because you defined it).
That points to an answer to your question: We compute residual deviance from the model by taking the approximation serious, that is, by using the deviance (-twice log likelihood, offset by some constant) calculated from the poisson quasi likelihood.
You can investigate this (in R) by simulating some (integer) data, then fitting a poisson glm, then a quasipoisson glm, and look for differences. You will see the deviances given are exactly the same.