This is what I have never understood. Why do you guys wait until after you graduate? Most firms hire new people from Sept.-December but again, many hire on a year-round basis. Since you have a degree in accounting, it should not be too difficult getting a job.
Contact your school's career services. Get in touch with alumni from school. And, look at several job databases and send your resume to hundreds of them. Make sure to write a nice cover letter. If you did get interview calls (not all will respond, so don't be frustrated if you don't hear back), then work hard to make sure you give your best. Prepare well for interviews.

And, H-1 is a lottery thing. So, there is no best way but rather just one way-find a firm that will sponsor for it and wish that your name will be selected. They don't have a quota for certain non-profits and academic institutions though.