There is no automatic update feature inside Blender. But getting the latest version is very simple - just go to the download page of Blender official site, download and install latest version.
Another easy method is to click "Blender Website" under the "links" section on the application splash screen.
From some time Blender are in Valve Steam. You can install Blender via Steam, which has a built in auto-update feature. This should keep your Blender up to date automatically when Steam worked. I.e. you can to get automatic updates for Blender throug Steam.
If you have installed Blender using Snap packages on a Linux machine.
To install using Snap you can use the following command on terminal:
sudo snap install blender --classic
Note: We need to have 'snapd' package already installed.
And, to update Blender using Snap you can you the following command:
sudo snap update blender --classic
If you're on a Mac and using Homebrew you can install/update Blender via terminal:
brew cask install blender
brew cask upgrade blender
Upgrade (all casks):
brew cask upgrade