This is a several part process:
0) You will probably need to DC restore the incoming video since it's probably AC coupled.
1) You need to demodulate the analog QAM modulated chroma information to recover the I and Q channels of the YIQ video.
2) You need to convert colorspaces from YIQ (roughly YPbPr or YUV - how far off it is varies with whether you're running PAL, "true" NTSC, or "fake" NTSC which is more common than "true" NTSC).
3) You need to recover the sync information from the incoming composite video.
4) (optional) You may want to separate the sync as PC monitors generally expect separate sync but often work with composite, but see step 5 since it may need one or the other
5) You need to scale the video from 480i to 480p (or 576i to 576p in 50Hz land) as PC monitors cannot display 15kHz scanrate video
6) Output to the PC monitor
Step 5 has to be done digitally to be practical and is usually done with a dedicated scaler chip or a DSP. You could also use an FPGA. There are chips out there that combine steps 0-3 for you from companies like TI or Analog Devices. You want what is commonly called a "video decoder". Step 6 would then be a video DAC and possibly an output buffer to drive the 75ohm input impedance of the PC monitor.
All in all, not generally a feasible hobbyist project. It's probably cheaper to just buy a box that'll do it for you or use an actual television.
If you happen to have a PC monitor capable of displaying 15kHz scanrates (rare, but they do exist), and you're OK with black and white video, just hook the composite video up to all three inputs and put the monitor in SoG mode. If the monitor can't handle SoG, use an LM1881 or similar to recover a discrete composite sync signal.
If you have a PC you can dedicate to this, you could use a TV tuner card. However, even in overlay mode, these exhibit some latency.