One way is to use the OUTPUT statement, with the needed keywords. You will need to check out the documentation, but something like:
output out=results pred=yhat resdev=resdev stdresdev=sresdev;
This example creates a file with yhat for the predicted log link, resdev for the deviance residual, and sresdev for the standardized deviance residuals. There are several other possible variables that can be created (other types of residuals, and so on). The file automatically has all your other (original) variables. Use a proc print data=results; to see what is in the file.
Then you can then use proc gplot, or one of the statistical graphics procedures to plot the relevant residuals vs. your predictor variables.
There may be other ods graphics options built right into GENMOD for doing this, but you will need to check out the documentation.