How I started my blog to get ARCs. I went to the book store and searched out books I liked. Wrote down the author's name and book, and googled them. If the author had a website I was able to contact them and ask that way. After awhile you get a reputation and publicists will start to email you to ask. Also this site has a publicist that asks you if you are interested in doing blog tours and reading their books.
Exactly what they said. I usually like to advertise their books somehow on your blog first and then email the author and be very polite and they'll tell you yes or no.
The most professional way is to contact the publisher. This way you can get the ARC's and be put on the publisher's list of possible reviewers for future copies. If you grow a relationship with a pub, they will start sending you their authors' books or asking if you'd like to review without you having to ask. This takes time to build though but, it's all about making contacts.
I can speak for many authors that most of us don't mind if a potential reviewer asks us for an ARC and are happy to be approached but some are not. Some authors frown upon reviewers contacting them straight. It's best to always go through their publisher. Also, the publisher will have ARC's before the author does. The pub will get it out to you much faster. Another thing is that you might not get in touch with the author in time. It takes some of us a long time to respond and you don't want to get lost in the shuffle so, contact the publisher with a professional email and they should assist you. Some might not contact you back but you can always keep nudging if you're interested in a certain title.
Another great way to get ARC's is by contacting the author's publicist whether it's their personal one or an in-house publicist. Publicists get ARC's out very fast for their authors.