Hello. There are multiple parts to this workflow.
First, you must create a network dataset, and in this network dataset, you must create an impedance (cost) attribute. It sounds like you already have a field in your street data where you have precalculated the impedance, so in your network dataset, you can just set up a field evaluator to read the values from that field.
Next, you need to create a travel mode in the network dataset that uses the impedance attribute you created. A travel mode is a collection of network dataset settings, including the impedance attribute, restrictions to apply, etc.
Finally, when you do your Closest Facility analysis, you need to select the travel mode you created using your impedance attribute.
You should probably start with this tutorial: https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/how-to-create-a-usable-network-datas... This teaches you how to create a network dataset and set up cost attributes and travel modes and has links to other relevant documentation.
After that, you can run the Closest Facility tutorial: https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/closest-facility-tutorial.htm
