After entering the US on a K1 Fiance Visa, you can apply for permission to work. You must file form I-765, Application for Employment Authorization. Once approved, you will be authorized to work for only a period of 90 days from the date that you entered the US.
A better approach is to file for your work authorization at the same time you file for your Adjustment of Status to become a Legal Permanent Resident. Under this approach, you would file your I-765 along with your I-485, Application to Adjust Status to legal Permanent Residence (AOS). There are two main benefits to waiting and filing your work authorization with your AOS. First, if you file your I-765 with your I-485, you do not have to pay the nearly $600 filing fee typically charged for obtaining your work authorization. Second, once approved, your permission to work is permanent so long as you are a Legal Permanent Resident.
If you have questions about the K1 Fiance Visa, please feel free to give me a call toll free at 1-888-515-3529.