How can I read the current from a monitor photodiode of a laser diode?

The Photodiode needs to be in reverse mode, it is wired for that correctly already (+12 V at the kathode. The anode just needs to be at a voltage that is below 12 V.

When in reverse mode, a photodiode will output a current that is proportional to the amount of light. To measure a current you need to connect a current meter in series. So just replace the voltmeter with a current meter.

The way you connected the voltmeter makes little sense as a voltage should be measured in parallel with a device, not in series like shown. (You could argue that you're measuring Vsupply - Vdiode but as the diode is in reverse mode, that does not make sense).

What you have works !
The diode is reverse biased thru impedance of voltmeter to measure current in high impedance.

For improved BW or speed, use a TIA with lower R.

fWIW your schematic in upside down. Use ground at the bottom with V1- = 0V = GND