D:OS Classic can run under XP, assuming the rest of the requirements are met, but the Enhanced Edition requires at least Windows 7 with Service Pack 1 installed, 64 bit version. The EE also requires a graphics card with hardware support for DirectX 11, which is unlikely if you have not upgraded it at some point.
Check your system specs. The EE is 64 bit, and requires a 64 bit OS to run. In the System Control Panel applet, General tab, check the CPU listed (if the computer came with XP, likely the CPU is 32 bit). Actually, you would get more detail by running dxdiag and checking the System tab (Click Start and then Run, or WinKey-R, type in dxdiag and hit enter).
If Windows is the only issue, the easiest way to upgrade would be to buy Windows 10 and install it to a new hard drive. That way you could dual boot for a while, until you had everything you needed moved over to the Win 10 install (email, bookmarks, etc). You should be able to find detailed guides on this process if you search online.
If you are comfortable dual booting, you could also consider installing Linux.
If you have a 32 bit CPU, the easiest way to upgrade is to buy a new computer. Technically you could replace the motherboard, CPU and RAM, then upgrade to a 64 bit version of Windows, but especially if you also need to upgrade the GPU (and you should check the power supply, as well, to see if it can handle the increased requirements), that is getting pretty close to the cost of a new computer.