Few things I can think of:
1. Verify IP settings - Assign the same IP/mask/gateway details to another machine and verify outside network is accessible
2. DNS issues - see if you can ping a global DNS server, 4.2.2.2. If this works, then the issue is with DNS
3. Virtual connectivity: verify the vmnic of the management vmkernel port. Ensure that the vmknic is connected to the same vmnic where you have external connectivity, also ensure that vlan and mtu details of the vswitch are proper
4. Physical connectivity: If possible, remove the cable from your ESX, connect to a PC, set the IP details and ensure outside network is accessible
If none of these are able to isolate the issues, then it could be that your server or NIC specifically is not supported by ESXi 5.1. Verify the HCL for this.