Step by step guide on upgrading a SharePoint 2010 site to a SharePoint 2013

1. Add and deploy any required O14 custom solutions in your SharePoint 2013 farm, notice that all O14 solutions being deployed under 14 hive but eventually it’s recommended to migrate the custom code to O15.
2. Backup and restore the SharePoint 2010 content database to your SQL Server hosting SharePoint 2013 farm.
3. Run Test-SPContentDatabase cmdlet to identify missing components along with potential errors and related warnings. Check the upgrade log and deploy any missing components and re run the cmdlet to verify.
4. Attach the content database to the desired web application using Mount-SPContentDatabase cmdlet.
5. After successfully mounting the content database to web application, the site should be accessible in 14 mode.
6. To upgrade the site simply click on “Start now” link on the toolbar, you can also go to SiteUpgrade page from Site Setting page as well.
7. Click on “TRY A DEMO UPGRADE” link to verify the site collection upgrade (This step can be performed by Site Collection Administrators, farm admins can run
   Request-SPUpgradeEvaluationSiteCollection cmdlet as well).
8. Provision a temporary site collection to validate the site post-upgrade.
9. An email should be sent out to Site Collection admin when the temporary site is provisioned.
10. After validating the temporary site we upgrade the site by going to SiteUpgrade.aspx page and clicking on “Upgrade the Site Collection” button (Farm admins can run Upgrade-SPSite cmdlet as well).
11. During the upgrade The SiteUpgrade.aspx page shows the progress and provides a link to an upgrade log for troubleshooting purposes.
12. Your site collection should now be accessible in 15 mode along with all new capabilities enabled in the farm and for the hosting web application such as Social and Office Web Apps.
    

Note: If there are any reference you manually added to CSS, JS, Images or any other files in your custom solutions which are in Layouts, you need to change the path of these references to point it to 15 hive layout.
e.g.
14 Hive – _/layouts/myPlugin.js
15 Hive – _/layouts/15/myPlugin.js