Since all my watched threads and PM's are set to send email notifications, I first check my email for new posts and things (this is convenient, because it means that if I don't get through everything then the notifications will still be waiting for me on my email and I won't "break" my alerts :P ). I often look at RP-related stuff and other important things first, then look at replies for everything else. If I'm online from my phone, or just not in a state of mind where I'd be able to give a good answer to something important, then I'll mark the email as unread and come back to it later (again, another advantage over on-site alerts :P ).
Once I've looked at everything in my email, then I go through on-site alerts. And because all the new posts I've gotten alerted about have already been checked through my email, the only things that actually show up as new alerts are just notifications about receiving ratings on my posts, and occasionally being tagged in things.
If I'm still not in focus-mode, then I'll just derp around and look at new replies to GC threads, and other discussion-based areas of the forum.
When I am in focus-mode, I'll go back and look at any RP-related stuff that I may have marked as unread and left alone earlier -- which typically means answering players' questions or updating my off-site posting notes to accurately reflect who most recently posted and who needs to post next. If I'm just poking around at stuff and don't have a ton of time to dedicate to RP's, then I'll go about doing maintenance duties, answering questions in the help desk, and replying to intro threads. If I am in the mood to do RP stuff, then, well, first of all, I'll probably have an IC post that I already wrote while I was offline and multitasking on other stuff -- so I can go ahead and copy that in and post it. Then I can send out a wave of posting reminders and attend to other miscellaneous GM stuff, and then maybe write a few more posts if I have the time (and spoons).