To update ESXi 5, you can do this from the command line or the GUI. This article teaches you how to update (patch) ESXi 5 and associate components using the GUI (VMware Update Manager via the vSphere Client).
How To Install VMware Update Manager
To patch ESXi, first thing to do is install the VMware Update Manager, which you can obtain from the vSphere install media or download from VMware.com as part of the full package containing vCenter. I like to install Update Manager on my vCenter server, because to me this seems like a good place for it.
Once you’ve installed VMware Update Manager, start the vSphere Client and click Plug-ins -> Manage Plug-ins. You will now see that there is a new plug-in available called VMware vSphere Update Manager Extension. Click the button to download / install.
This is how to install VMware Update Manager, once this is complete you are ready to update ESXi 5 and other vSphere components though it.
Updating ESXi 5 Hosts
To use VMware Update Manager, in the vSphere Client, navigate to Home -> Solutions and Applications -> Update Manager. On the Getting Started page, click Download patches and upgrades. Next click on Configuration and configure the update settings on the left hand side. At the top right hand side, switch from Admin View to Compliance View. Select the host you want to apply updates to on the left hand pane. Right click the host and put it into maintenance mode then click on the Update Manager tab. Right click in the area for Attached Baselines and attach the baselines that you want your hosts to be patch to (the default two are fine).
Now click the Stage button and follow the next buttons to download patches ready to update your ESXi 5 host. Once these have finished downloading, click the Remediate button to apply the downloaded patches to your ESXi host.
Reboot the ESXi host and take it out of maintenance mode. You have now successfully patched your ESXi 5 host.
You may get for following error after you click Remediate:
The operation is not supported on the selected inventory objects. Check the events for the objects selected for the operations
To fix this, see this article: VMware Update Manager – Operation Not Supported.