You can either rig the old hard drive into your new machine and then format it or you can buy an external USB enclosure and format it from there. Rigging it into your rebuilt computer is the cheapest option though.
Many Format operations do NOT wipe out old data except for the small area used by File System files. Ideally what you want is to Zero-Fill all of your old HDD, then Partition and Format it.
Is that old HDD made by Seagate (or Maxtor) or Western Digital. Each of them has a free downloadable disk diagnostics utility you can use on their drives. Seagate's is Seatools (works on Seagate and Maxtor), WD's is Data Lifeguard. Each has a Zero Fill tool you can use to really wipe an entire disk clean. If your HDD is by another maker, check their website for such a utility.
Does the mobo have built-in video? If so, it may be using that as output, and not your video card. Some such systems need you to set the BIOS to use an add-in card. If that might be your case, try feeding the monitor from the mobo VGA output connector, not from the video card.
The other thing that can give you no performance is if you fail to connect the power cord from the PSU to the special 4-pin connector near the CPU. Some mobo's have this one is in addition to the main 20-pin power supply connector elsewhere.