Generally speaking, you must be the person's personal representative. Since you were not legally married, the best option for personal representative in this case would be legal next of kin (a parent, sibling, or adult biological child). An unrelated person can be a personal representative for income tax purposes, but it would be best to go to the probate court and ask for an appointment as the person's executor. For a small estate, this may not be complicated; you might only need to file a form, accept responsibility, and have the agreement of his legal next of kin that it is ok for you to do this. You might want to check with the court, they probably have a web site, or a clerk's office that can assist you with basic information that applies in your state.
https://www.irs.gov/forms-pubs/about-publication-559
If you meet the definition of personal representative in publication 559, and no one more closely related objects to your doing this, you may be able to take this role without filing any forms with the court. but it really depends on the law of your state and how his family feels, and you may want legal advice.
Also, because you were not married, it is possible that the legal rights to his property (bank account, car, furniture, investments, anything else) go to his next of kin and not you, unless they don't object. If you have a child together, the child might inherit his property, with you there to "help". Again, you may want some legal advice.
