These symptoms do seem to be a GPU issue. As a stopgap you can use your PCs integrated graphics (sometimes called an iGPU or APU, if no image appears after following these steps, your PC may not have an iGPU present). If you are comfortable of removing the part yourself, disconnect your PC from its' cables, open the side panel and ground yourself by touching the sides of the metal case, making sure that lose clothing does not make contact with the components.
Locate your GPU, from the back of the PC unscrew and remove any bracket holding the GPU in place. Then from the inside of the case look at the back of the connection between the GPU and motherboard, pull slightly back on any retention mechanism (usually a plastic strip) and then pull the card upwards, trying to avoid the individual components on the GPU board if possible. Note that relying on integrated graphics will severely reduce performance in games and some other applications. Changing to your integrated graphics may temporarily lower your display quality until Windows locates drivers.
Reconnect the PC, connecting your monitor cable to the blue VGA port on the motherboard and boot the computer. You can check your RAM through Windows itself. Press the Windows Key + R, type “mdsched.exe” into the Run dialog that appears, and press Enter, Windows will ask to restart to complete this scan. Motherboard issues are less likely, however you should check the instructions for either your motherboards specific model or from your computer manufacturer if provided, as each motherboard is slightly different. You may also chose to check the health of your Hard Disk Drive (HDD) by using a program such as CrystalDiskInfo to ensure this is healthy. If the symptoms persist, you may need to look at your CPU before ordering a new GPU.
You can save money on a new GPU by buying on the used market. Certain older cards, such as the AMD Radeon HD 5670 are still fairly capable today for their prices, even though they will no longer receive new drivers. When you do replace your GPU, ensure that the wattage of your power supply is sufficient, you can check this with EVGA Power Meter. You should also check if your specific card requires a dedicated power cable, as some power supplies lack these.