Excellent detailed post: Live Migration of SharePoint 2010 to SharePoint 2013
This post will give you a step by step process on how you upgrade your SharePoint 2010 to SharePoint 2013. While this is a live migration I suggest you do proper backups of the server and the database before continuing. Having said that live migration means that you can upgrade your SharePoint instance from one server to another without any downtime, but during the process anything that happens between like adding new items on the list or deleting a wiki page will not be captured after the copy database has been performed.
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