Japan-Career.jp is a career consulting firm based in central Harajuku. They specifically cater to foreign nationals looking for work in Japan. The jobs range from small ventures to large Japanese companies, often in sales and marketing divisions.
These are primarily 正社員 positions aimed at the under-35 crowd. Some positions require an existing skillset, whereas many others are meant for recent or semi-recent college graduates who can fit the Japanese concept of 新卒 or 第二新卒.
As for qualifications? You need to have excellent Japanese. JLPT-1 or a high score on BJT, and speaking ability to match.
However, if you meet these requirements, their hookups are good. They regularly pick up contracts from big players like DeNA, Takeda Pharmaceuticals, Daiwa Securities, and Square Enix.
The best way to work with Japan-Career is to call or e-mail them and schedule an in-person interview at their offices. The building is about a 5 minute walk from Harajuku Station. The facilities are small, but nice, and the people there give great individual attention. They help job-seekers identify job openings in which they'll be a competitive candidate, and may introduce you to industries and fields you hadn't considered before.
And, even if you don't find a listing with them that appeals to you, their website is a great source of information about resume preparation, interviewing skills, and stories of other job seekers. You'll especially want to check out their magazine, also titled JapanCareer. PDFs of the latest editions are available for free download at their homepage.
Mind you, all of this is free. Since the hiring companies pay for consultants to identify and direct qualified applicants to them, job-seekers do not pay anything for these services.