Getting married under Japanese law is comparatively easy. Unless a couple is in violation of bigamy, consanguineous marriage (i.e., marriage between close relatives), or within the period of prohibition of remarriage (i.e., basically 100 days for women after the divorce of the previous marriage), you just have to file a registration paper to marry someone. There is no need to hold a marriage ceremony like in other countries.
If you will have a marriage ceremony in Japan and one of the couple is a Japanese, Japanese laws will govern how you may get married. In other words, you just have to fill out the registration paper (婚姻届) in Japanese and file it with the municipal office where your residence is located.
With the registration paper, you, if you are a foreigner, also must submit a certificate of legal capacity to marry someone, more simply put, a “single status certificate.” You can get it from your embassy or consulate. It is a substitute for a copy of a certificate of one’s family register (戸籍謄本) which is required for a Japanese to get married.
With marriage, you will be admitted a resident status as a “spouse or child of a Japanese national” if you have a valid status at the time of the marriage. The period of stay will be 5 years, 3 years, 1 year or 6 months. (See here for details.) To acquire the status, you have to apply for the change of the type of resident status.
Even if you do not have a valid resident status, you may get married if you can submit a single status certificate. You can apply for a resident status as a “spouse or child of a Japanese national” in the deportation procedure by obtaining a special permission of stay (特別在留許可) from the Minister of Justice (法務大臣).