To confirm, try pinging Google's DNS: 8.8.8.8 If that works try manually setting the nameserver.
I wouldn't imagine that having 2 NICs would cause any problem. I haven't installed TKL to hardware at all but I have used many different versions of Debian & Ubuntu with multiple NICs and haven't had any issues like this, although I have had some minor issues getting them both to work.
I would suspect DNS issues
To confirm, try pinging Google's DNS: 8.8.8.8 If that works try manually setting the nameserver.
I wouldn't imagine that having 2 NICs would cause any problem. I haven't installed TKL to hardware at all but I have used many different versions of Debian & Ubuntu with multiple NICs and haven't had any issues like this, although I have had some minor issues getting them both to work.