Your going to want to enable IP forwarding, and then either setup your route tables or use IP tables for NAT'ing. Here is information on how to set up NAT'ing:
echo 1 > /proc/sys/net/ipv4/ip_forward
/etc/sysctl.conf and change the line that says
net.ipv4.ip_forward = 0 to
net.ipv4.ip_forward = 1
/sbin/iptables -t nat -A POSTROUTING -o eth0 -j MASQUERADE
/sbin/iptables -A FORWARD -i eth0 -o eth1 -m state --state RELATED,ESTABLISHED -j ACCEPT
/sbin/iptables -A FORWARD -i eth1 -o eth0 -j ACCEPT
I believe you can just drop the MASQUERADE line if you want forwarding and no NAT.
Other than your specific example of an HTTP request, the rest of your details indicate you want an actual gateway, and not just a proxy.