PC 12v rail shows ripple under load when checked through a multimeter
I am wondering if this is the right way to check for ripple in PSU?

I would say no unless you know the behavior of your DMM. On most DMMs the AC range is designed for low frequency (often up to 400 Hz) and sinusoidal signals.

How will your DMM respond to high frequency AC, your PC's power supply will output DC rails with a (for example) 100 kHz ripple due to the switching of the DCDC converter in that power supply?

How will your DMM respond to a DC voltage being present while measuring AC?

If that ripple was indeed 100 mV to 200 mV that is to be expected and nothing to worry about. Most circuits in a PC are designed to allow for so much supply ripple.

Also while the 12v rail drops to 11.6v, the 5v rail increases from 5.04v to 5.15v

Also nothing to worry about. In many PSUs (Power Supply Units) for PCs the 12 V and 5 V outputs do not have their own regulation loop but share one. So it might be that the 5 V isn't regulated by itself but relies on the 12 V rail.

If you load the 12 V rail the PSU has to "work harder" to compensate, that might increase the voltage at the 5 V output which might still have the same load.

The drop from 12 V to 11.6 V can also be (partly) due to IR-drop, meaning voltage drop through the wires and connectors.