If you have in mind a particular publisher's list, get the contact details for the person in charge of the list you want your book to be included in. Ring them or send a brief courteous email asking if they still run the list and whether they would welcome a proposal. Otherwise, contact the publisher's commissioning editor for the appropriate area (you may be able to find their name and contact details on the website). Check what they expect you to send (commonly a covering letter, a book proposal and a sample chapter or two) and whether they prefer it in hard copy or electronic format.
Approaching more than one publisher at a time
Don't do it! You may think you are speeding up what can be a very lengthy procedure (see 'How long will it take?') and also doubling your chances of acceptance as you sharpen the competitive edge between publishers, but the reality is that they will probably be annoyed and both drop you. There is not much point hiding the fact: either the publishers will know each other or the readers will know each other, or your material might even be sent to the same reader! In all events, it tends to be very much discouraged by the industry, so if you are going to do it, tell them. This is even more important with articles since journal editors and academic reviewers do not get paid for the time they devote to refereeing your article.
Sending the proposal:
The covering letter
This should be brief, direct and clear, introducing yourself and the work you are sending. All main points will be covered in the book proposal.
The cook proposal
This document should not be longer than 2 or 3 pages and should include the following information (courtesy of The Nordic Institute of Asian Studies website):
Sample material (see also The writing process)
If they ask for a sample chapter or chapters, these 'will be used by the publisher to assess the author's command of both subject and style (essentially, to judge how much work will be required to make the manuscript publishable)' (Nordic Institute of Asian Studies). Your material may not be sent at this point to a reader or readers, but is more likely to be looked at by the publishers themselves as they decide whether it is publishable and whether it will fit their profile. Spend some time considering which chapter will be the best to send – an introductory one tends to be the best as it is most approachable, but you may feel you prefer to send a particular study, which shows in detail the kind of work you are doing. You may be able to send more than one. Consult with your supervisor about this.
Even at this stage, the publisher may ask for the opinion of expert readers; this is all the more likely if they asked you to submit a whole MS. Make sure that the work you send is properly proofed, well-presented and preferably also formatted in accordance with the publisher's style (Chicago, MLA etc).