To create the Customers as Community Users you would not normally need to use a permission set as you have to have profiles created for the community users and assign that profile to the Community. It is possible if you have multiple communities to use a permission set rather than the community user profile, but if you only want to have a single community it is easier to do it from the profiles.
Then the users need to be created from the Contact records, this can be done manually from the individual contact records. Once enabled these users can be added to a permission set for their profile type as normal, e.g. Customer Community or Customer Community Plus.
To enable a community user from the Contact select Manage External user and then enable customer user. This takes you to the user creation page where you can enter their details, as per the next two pictures.
The role is not available on community user licences and is defaulted in on the customer community plus licences. The Profile has to be for the specific licence type and needs to be included in the community profiles. Once these users have been created then it is possible to assign them to a permission set but that may not be necessary as explained previously.
See also Salesforce documentation on provisioning communities users.