Can I ask the editor for rapid processing (rapid peer review) in the cover letter of my paper submission? I'm a PhD student and I need to (almost desperately) squeeze in another paper for my dissertation in the next 6 months. Is it acceptable to explain my situation to the editor and ask to set a tighter deadline to the reviewers?

My field is health sciences and biostatistics.
You can ask for just about anything, but unless you already have some connection to the editor you aren't likely to have much happen out of the ordinary. In particular, they are unlikely to move you ahead of another author and you would probably object if the situation were reversed. 

The editor has little control over what his/her reviewers do or their schedules.  If he/she tries to press them, they might just turn the paper back and you get in line again. The editor is unlikely to move on a paper without reviews. The time it takes is the time it takes. 

Note that in some ways the reviewer "stable" is more important to the editor than the current bunch of authors since there is a long term relationship between editors and reviewers. Since they probably aren't paid, there is little leverage. 

But you can ask.