Best way is to use your Multimeter.
I would look up the schematic diag or the parts list like this
https://www.centurytool.net/141_037_John_Deere_15_2_150_amp_Battery_Charger_s/21732.htm
It looks pretty generic.
Check output of transformer ... if transformer is two wire it's just AC vote about 18V. if it is 3 wire.. check the two voltages that is coming out, it should be reference to the center tap.. center tap is ground for these type of setup. You might encounter those multi multi tap transformers that dependents on the selector, different amp winding.I think... never got the hang of that thing.
Check the AC input of the diode (S), then check the DC output
Check the capacitor with your cap setting.... it should be relative close to the marking. Not sure if you have the capability to check ripple voltages.
Check the selector switch. some times these multi position switch gets oxides and acts up.
Check if there is a round Disc type look alike fuse on the Negative cable inside... and check if it is heating up when the charger exhibits this behavior... $5 or $7 item at Autozone. It's the automatic reset fuse..
I think that is all I remembered.. .maybe there is more if some one chime in. My HF one that looks like that was the problem with the auto reset fuse when there was no output.