hi,
The opto coupler is
essential for safety.
Have you wired the circuit
exactly as shown in your diagram.?
Which PC operating system are you using, eg: Windows 98, XP Pro etc.
To test the circuit it is not necessary to have mains connected to the relay contacts.
When the relay operates you will hear it 'clicking'.
As a simple test, remove the 470R connected to pin 4 of the opto to the transistor.
touch the free end of the 470R to +6V, the relay should operate.
If that worked OK, reconnect the 470R, then remove the 470R from pin 2 [D0] of the 25way connector,
connect the free end of the 470R to +6V, the relay should operate.
Do you follow OK.?