Since you say, that you have green bars when the vinyl is playing, then it sound like audio is arriving at the PC, so check the PC audio settings, for sliders down at minimum, muted devices etc.
Right Click the speaker icon in the system tray and select 'Open Volume Control' then in the windows that appears Options/ Properties, and click the 'Recording' Radio button.
Make sure that if you have multiple devices in the Mixer Device Drop down box, that the correct one is selected (you may need to experiment here if you have multiple devices).
Also ensure that there is a tick next to the 'Line In' volume control.
Click OK and select Line In as the input channel.
Now right click again on the speaker icon in the system tray and then select 'Adjust audio Properties, then select the Audio tab, and check that the Sound recording device is set to the correct device as setup earlier
Just to be sure, also double check your setup and simplify it as much as possible. Even to the stage of getting it recording correctly first with no audio output, then after get audio working on the output side. If you have a headphone socket on the front of the PC, try monitoring the audio first from there without using the amp for playback.
I have also read your setup, and can not fully understand how you have it setup. It sound like you have the Rega plugged into a Y cable / Splitter.
You also say you have a Y cable connected to the Blue Line in on the Dell.
If that is the case, then it will not work. You need to have a chain, with no Y cables/ splitters.
Chain should be.
Rega plugged into phone stage of the amp.
Tape Out from Amp into the Blue Line in on PC.
To listen to the audio from the PC through the amp, then you could connect the Line Out ( Green) from the PC back to a line in on the amp ( Tape in or any spare line in on the amp)
I also have a Rega Planar 3, with a Moving Coil cartridge, so the output level is even lower than a standard Moving magnet cartridge. If you have a MC cartridge on the Planar 3, check that the amp is set for MC and not MM.