One item that should be in every camper is a multimeter that will read voltage and resistance. They can be had for very little money but can save your butt on trip. If you are checking for the battery charging, then a simple voltage check will show it. Disconnect the battery and read the voltage across the 2 wires that you removed, if it shows around 14v DC then you are charging. If no voltage then the converter is not working and you should be looking at the power to the converter, first are you getting 110v AC from your AC pull in, if no, check the power panel at the post outside. If yes, check the breakers and fuses in the trailer. If you have 12v for your lights
with the battery disconnected then the converter is half working but not charging the battery. At that point you will have to get into the bottom half of the converter and check the load resistor and fuse on the back wall of the converter(may have been blown if the battery was connected backwards even for a second).
Having a multimeter is the first part, basic knowledge on how to use it is the second part. Lots of good websites that will show what settings to use and how to use them. Don't forget that there is AC and DC voltage in the trailer and if the meter is set to the wrong voltage type you will get wrong readings.