It is unfortunate to inform you that you can not access GCSS from your home computer. Your computer will still fail to access even though you have a CAC reader or installed the necessary software.
You need a government computer with a VPN to access GCSS from your home network to rephrase it.
You must fulfill one of these two requirements if you want to be able to access GCSS. You need either a .mil network connection or VPN capability to a .mil network.