If present, use the motherboard video out connector (bypassing any dedicated gfx card). You'll probably need to go into the bios at startup and change the default display accordingly.
Additionally, reconfirm pc to monitor lead/connections. Try replacing lead if you have a spare.
Most monitors will reconfigure to the input resolution (output from the pc), however, you might want to reconfirm the monitor settings.
Try resetting the bios ('load optimal defaults' - or similar wording, in the bios page). If you do this, note down any bios changes you required previously so that, if required, you can re-enter them later.
As another option, there is another way. Providing you can connect the old pc's HDD to the new pc (i.e. you have a compatible connector), then boot from the rescue disc of software such as Acronis Disk Director, or Paragon Partition Manager. Once inside this software you'll have all the copy functions you'll need.