How to write PIs asking for PhD position?
When drafting your email, keep in mind that PhD advisors get tons of request like this. Your request has to stand out in order to have a chance of success. Most email requests are pretty generic, simply asking for a position, with some generic praise of the advisor's reputation, and a CV attached. 

In order to stand out from the mass, the most important thing is probably that you do your research. What topic is the advisor working on? How does your background fit in there? Make it clear what you think you can contribute to the advisor's work. Try to get all this across as concisely as possible, maybe three or four sentences, at most. By all means, avoid being generic. 

Surely you will have to put some time into it, but it will increase your chances of being considered dramatically. Better to send out fewer but well targeted emails than masses of generic requests. 

If you get no response in spite of your request being well crafted, you can try following up once, after a week or so, but not more often. 

EDIT: "Doing your research" does not mean writing a research proposal, but finding out what the PI works on and putting your skills and interests in this context. Most of the time the info on their website is sufficient, you don't have to read every single one of their papers.