Using Local (private) DNS server on Lollipop I have a website hosted on an IIS server on my network. The DNS for this website on only valid from within the network. The DNS cannot be resolved outside of this local network.

I am using Wi-fi to connect to the network, but I believe my phone is trying to use the google dns servers in order to resolve the website name when I try to access it from chrome in my Galaxy S5. 

Are there any workarounds for directing android to use a seperate dns server ?
You can modify the network config to set a specific DNS value.  Find the WiFi network you are connected to, press and hold on the name of the netowrk->Modify Network Config->Show Advanced Options.  From here you can set a specific DNS, after changing the IP Settings to static.

I believe that most devices look at the DHCP for a DNS server, with a google IP as a backup.

There are some handy apps (must be purchased though) that allow you to only change your DNS but keep your IP address dynamic, based on DHCP.  Overrride DNS