K visas are meant for the fiancé(e)s of US citizens and their minor children. To sponsor their spouses, US citizens must file Form I-129F, Petition for Alien Fiancé(e), on behalf of their fiancé(e)s. The fiancé(e)s of US citizens who enter into America on K-1 visas can apply for adjustment of status but they can do that only after they get married to the US citizens who filed Form I-129F for them. If an immigrant petition was filed for you by a US citizen and if you had entered into America as a K-1 non-immigrant, you will become eligible for adjustment of status. Nevertheless, you can file Form I-485 for adjustment of status and obtain a Green Card, only after you get married to the US citizen who sponsored you and that must be done within 90 days of admission. You can apply for adjustment of status, if an immigrant visa is available and you need not worry about the availability of visas as the spouses of US citizens will not be put on waiting lists.
You need to marry your US citizen fiancé(e) within 90 days of entry into the country and you will become eligible for adjustment of status only after getting married. You will be placed in removal proceedings, if you fail to get married within 90 days and you cannot adjust your status through any other means. That is because, K-1 fiancé(e) visas are valid only for 90 days and the fiancé(e)s who get into the country on these K-1 visas will have to get married within that 90 day window, in order to remain in the country and to apply for adjustment of status. After getting married, you can file Form I-485 and change your non-immigrant status to immigrant status.
You need to submit Form G-325A, Biographic Information and Form I-864, Affidavit of Support along with Form I-485. Similarly, you may also file Form I-765, Application for Employment Authorization and Form I-131, Application for Travel Document, if you seek to work in America or travel abroad while your adjustment of status application is pending. You need to file Form I-485 along with a copy of your marriage certificate, Form I-94, Arrival/Departure record and the fiancé(e) visa petition approval notice. Similarly, you will also be required to appear for an interview and will be required to bring with you few documents to the interview. You may bring original documents and copies of those original documents as the USCIS officers may verify the original documents and keep the copies of those documents with them. After your adjustment of status application is adjudicated, you will be notified about the decision by mail and if your petition is approved, your permanent resident card will be mailed to you.