How can I set / change DNS using the command-prompt at windows 8

First, the network name is likely “Ethernet”, not “Local Area Connection”. To find out the name you can do this:

netsh interface show interface

Which will show the name under the “Interface Name” column (shown here in bold):

Admin State    State          Type             Interface Name
-------------------------------------------------------------------------
Enabled        Connected      Dedicated        Ethernet

Now you can change the primary dns (index=1), assuming that your interface is static (not using dhcp):

netsh interface ipv4 add dnsserver "Ethernet" address=192.168.x.x index=1

2018 Update – The command will work with either dnsserver (singular) or dnsservers (plural). The following example uses the latter and is valid as well:

netsh interface ipv4 add dnsservers "Ethernet" address=192.168.x.x index=1