How to manually upgrade your Office 365 SharePoint from 2010 to 2013 if you have an E1 or E3 plan

Login to the MOP as a global admin. Click on the Admin drop down on the far right by your name and click "SharePoint"

On the next page, you will see a list of all your site collections. Look at the last column and you will see what version of SharePoint that Site Collection is on. If its on 2010 and you want to upgrade it, just put a check in the box right before the URL, and on the top ribbon click "Upgrade"

When you click Upgrade, a small window will pop out from the bottom of that button and you want to select "Site Collection Upgrade Settings"

You will then have a window pop up. You want to click the link that says "Link to the upgrade page"

That will take you to your SharePoint upgrade page. All you need to do is click that blue box that says "Upgrade this Site Collection", and follow the on screen instructions. It will update automatically and once its finished, you will be on SharePoint 2013. You can go back to the Site Collections and do the next one. You only one to do one at a time.