When I picked up my very first Famiclone, a Power Joy without the battery door, I was completely ignorant of what power supply to use. The only thing I had around at the time was an NES AC adapter (NES-002), and instead of seeing anything light up on the screen I quickly smelled the melting of internal components. For years I thought I had burnt the thing to a crisp and it would never work again, later I found out I didn't do it any harm, luckily!
During the years that followed all I found were Super Joy 3 systems, which use a much easier to understand and more readily available battery power source, so it wasn't until I acquired a complete Super Joy 3 system that I ever really worried about a wall wart slowing me down. Having this one power supply subsequently helped me save tons on batteries and it also helped me test out my Super 57000 Video Game, as it was just the console and controllers, no power supply. But most importantly this one power supply ran all of my Famicom clones without any issues, but I needed more than just this one for my entire collection.
I now own 2 different Famiclone power supplies which show me the needs of a Chinese Famicom clone are quite simple to sustain. And although these power supplies are for 2 different styles of Famiclones, they work across the board on all of my consoles and handheld systems that offer the option of being powered by a wall wart. But as with anything from China these things are super cheap plastic and I wouldn't trust running my systems on them for both my own and the Famiclone's safety, but I'll get into that later.
Do not feed your system too much voltage, and do not mix AC with DC, all mine work on strictly DC voltage. You can run slightly higher amperage without worry as the system only takes what it needs, but do not starve it by using a power supply that offers less amperage (about 500mA is ideal). Last make sure you have the correct plug polarity, positive outside and negative inside, most power supplies have a diagram on them showing which polarity the output plug will give your device.
Since the supplied DC adapters for Famiclones are pretty standard Chinese junk, I've found a handful of alternatives that are substantially better quality but offer the needed power for my systems. I haven't tested them personally but I also believe a Sega MK-3025 or MK-1602 should work perfectly fine as well. In short make sure the power supply works with your wall outlets, provides 9 volts DC and has as much amperage as your Famiclone needs, and finally make sure what you plug into your console has outside positive and center negative polarity.
There are the simple rules I've learned and currently follow. My Famiclone collection is always growing and I'm not the type of person to turn down a good Famiclone simply because it doesn't have all the hookups needed. Now that I know what most Famiclones run on, I've got the most important part covered!