After "ifup" or "networking restart" command, you have to check your network interfaces with "ifconfig" or "ip addr" command. Also checking routes of device may be useful.
Sample "ifconfig eth0" output:
root@pi:~# ifconfig eth0
eth0 Link encap:Ethernet HWaddr b8:27:eb:30:42:ef
inet addr:192.168.1.5 Bcast:0.0.0.0 Mask:255.255.255.0
UP BROADCAST RUNNING MULTICAST MTU:1500 Metric:1
RX packets:243477 errors:0 dropped:0 overruns:0 frame:0
TX packets:102566 errors:0 dropped:0 overruns:0 carrier:0
collisions:0 txqueuelen:1000
RX bytes:197482011 (188.3 MiB) TX bytes:11086659 (10.5 MiB)
Sample "ip addr" output:
1: lo: <LOOPBACK,UP,LOWER_UP> mtu 16436 qdisc noqueue state UNKNOWN
link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
inet 127.0.0.1/8 scope host lo
2: eth0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc pfifo_fast state UP qlen 1000
link/ether b8:27:eb:30:42:ef brd ff:ff:ff:ff:ff:ff
inet 192.168.1.5/24 scope global eth0
Sample "route -n" output:
root@pi:~# route -n
Kernel IP routing table
Destination Gateway Genmask Flags Metric Ref Use Iface
0.0.0.0 192.168.1.1 0.0.0.0 UG 0 0 0 eth0
192.168.1.0 0.0.0.0 255.255.255.0 U 0 0 0 eth0
To connect to internet through "eth0" interface, you should see the line with "UG" flag correctly. That line means that every packet from the local network will go to outside through 192.168.1.1 and eth0 interface. Without this line device can't know where to go.
If there is no route line which has 0.0.0.0 destination, you can test with adding by manually with "route command" ;
route add default gw 192.168.1.1 netmask 255.255.255.0 eth0
Also check your DNS settings by "/etc/resolv.conf" file. Because you not mentioned how you test the internet connection maybe your system does not resolve addresses. Nameserver line should contains a DNS server IP address. To test it you can use Google's DNS server IP addresses. (8.8.8.8 or 8.8.4.4)
root@pi:~# cat /etc/resolv.conf
nameserver 192.168.1.1