To upgrade a Site collection from SP2010 to SP2013 we can only utilize the Database Attach approach. The In-Place upgrade method is not available with this version.
Below is the link for 6 part videos explaining the process in detail:
1. SharePoint 2013 Upgrade: Overview
2. Creating a SharePoint 2013 Farm for the upgrade:
3. Copying SP2010 Database to SP2013 Farm:
4. Upgrading the Service Applications Databases to SP2013:
5. Upgrading the Content Database to SP2013
6. Upgrading the Site Collection to SharePoint 2013