For inverters and inverter chargers, first inspect the physical condition of the unit to look for any possible irreparable damage. Typically you begin by connecting the inverter device to a battery. You also need to check with the multimeter at the inverter terminals when connected to the battery if its the same voltage reading as from the battery terminals. Afterwards, the AC output is typically tested using appropriately sized loads based on the rated watts of the inverter. Check for any fault or warning lights when doing this. Also check if the fans are too noisy or overworked, and if the unit is overheating.
If the unit is an inverter charger, you must also test if it can receive shore power by connecting the proper live, neutral, and ground (L,N,G) wires to the correct input terminals, as well as checking to see if it is charging the battery.