Hello,
Yes this is possible. Your PC can have default gateway to the router.. the router can route only the required private subnet over GRE. by doing this, u can access both your remote private subnet and internet.
GRE is easy to configure, but i would never use it, since it sends packets over internet on clear text.. it is not at all secure.. i would instead use IPSEC with 3DES encryption, on a client-server/remote access mode. In this, the client will connect to the remote end VPN server (can be a router/firewall/ VPN concetrator etc).. Once authenticated and connected, he can access any resource on the remote network. If we enable something called "split tunneling" he can access both remote private network as well as internet on the same PC. Only the traffic to the remote office IP will go through the VPN client adaptor. all other traffic will go through the physical NIC, thus allowing both the accesses.
Hope this helps.. all the best.. rate replies if found useful.
Raj