Replace VLAN with network and read this: http://serverfault.com/questions/49765/how-does-ipv4-subnetting-work
You have a network 192.168.0/24. All traffic on for that network gets sent to it. All traffic with a different destination get send via the gateway 192.168.0.1, which is on the 192.168.0/24 network and thus is directly reachable.
Now you add a second network. 192.168.5/24. All traffic for that specific network can be sent to it. All traffic with a different destination needs a gateway which is directly reachable from 192.168.5/24. That does not include 192.168.0.1.
How would adding a default gateway to 192.168.5.1 to allow me to get internet access?
It would not. You will need a NIC in the local network (192.168.5/24) which can forward packets to and from other networks. Assuming that everywhere you wrote VLAN you meant 'network' or 'LAN' then you can do this by adding a second IP and appropriate routing tables to the host which currently serves as the gateway for the 192.168.0/24 network.
However as your question currently stands that requires some guessing. You will either need to clarify it or learn how subnetting and routing works.