I upgraded my SharePoint Server to the latest June update today. It is a massive hotfix download 1.8GB.
The install is pretty straight forward, after you have downloaded the file you extract it to a folder.
I normally open SharePoint Powershell as Administrator and then navigate to the folder where I extracted the CU.
Installing the update takes about 2-3 hours, obviously depending on you hardware. After that you need to run the SharePoint Product Configuration Wizard to actually upgrade to this update.
Do take note that you cannot install the June CU if you have not applied the March CU.
After the update has completed and all services are started your site should be operational once again.
Hope it helps