If you have a DVI to DisplayPort cable and it's not working when connected to your graphics card and monitor, it's unlikely to work when connected to your CPU.
The reason for this is that the DVI to DisplayPort cable is designed to convert the DVI output from your graphics card to DisplayPort input on your monitor. Your CPU does not have a DVI output, so there would be no signal to convert.
In general, it's best to connect your monitor directly to your graphics card for the best performance and compatibility. If your graphics card does not have a DisplayPort output, you could consider upgrading to a newer graphics card that supports DisplayPort or using an adapter that is specifically designed for your graphics card and monitor combination.