Well... if you're here, then something must have gone wrong. Damn...
Do not despair, take a deep breath and start working through the basics. A frequent problem that has been posted on the site for the Pi1541 firmware itself, is that under-powered power supplies will often cause problems for you. While some may consider it a design weakness, I've found that when it comes to powering a Raspberry Pi 3, or later versions for that matter, there are no easy solutions beyond plugging a PSU straight into it - and so it takes a PSU providing 5V DC via a barrel plug. In todays market there are many power supplies that are simply unbelievably cheap, the last one I bought started gradually toasting whatever was plugged into it after a year or so. One affordable, yet trustworthy PSU is the GST25B05-P1J produced by Mean Well (this has a EU plug on it, for other regions you may need a slightly different model).
With the power issues either solved or ruled out, does the display work? If it is completely dark, take the Pi out of the Pi1541-III and allow it to start up with HDMI connected to a modern monitor. If you either get a black screen or a rainbow-coloured one, please check your SD-card - can you tell if there is a difference when starting up with or without a card inserted? The most likely cause of your problems is that the card has been formatted using the exFAT file system from the factory, a Raspberry Pi is currently unable to boot from any other file system than FAT32. If that is the case you need to format the card to FAT32, something that isn't straight forward on newer versions of MS Windows - one third-party tool such as fat32format can format the card for you (just don't accidentally format a different drive by accident).
Put it back in again, still nothing on the OLED-display despite having checked that the configuration is correct? Double-check your previous notes so that you've soldered JP3/JP4 on the Pi1541-III Module Panel according to the actual pin order on your OLED - if you've forgotten to do so, then do it now and try again. Did you do it the wrong way around and have already attempted to power up the thing, you know that it is probably dead now? In which case, you should order a new display though you may have gotten lucky. If you observe graphical glitches, you may have configured the display as a ssd1306-based display and although the sales listing usually say ssd1306 - every 1.3" display I've encountered are actually based on the sh1106 and need to be configured as such.
You've gotten this far, then that's super-awesome! Hopefully you can select a d64 file from your SD-card, move up/down to the file and click select to start emulating a drive with that disk inserted. Try to load up the directory listing from your Commodore 64:
LOAD "$",8
If it fails instantly with a message saying "DEVICE NOT PRESENT", your 7406 might not be working as it should or possibly not at all (ensure your C64 actually works with the same cable and with a real disk drive first, if possible). Try a different chip and hope that it works.
If you get a different error or quite possibly the Commodore 64 just hangs there without doing anything, a good bet is that the I2C level converter is crappy - I've had around 20% DOA on those things, so it is not unlikely. My best recommendation is to use a multimeter and measure the resistance across various points, more than likely you can spot the one where nothing is connected by comparing between them (they usually come in packs of 5 or 10 so you should have spares).
Something else that may come up is that some 7406 chips have been distributed with known issues, these can cause slightly less dramatic faults and simply cause data to load in with errors. For this reason the creator of the Pi1541-software, also recommends that you test by loading in Ghost'n Goblins Arcade to check that the chip is performing adequately - in particular the loading screen will have graphical issues. No Pi1541-III is therefore never considered adequately tested before it can reliably load this particular disk image. Your mileage may vary, but the chips that I've had the most luck with have been specifically marked 7406PC and can be usually found for sale at a higher price than the cheap ones.
Finally, if the computer just says READY then that is not an error - you just need to brush up on your BASIC commands. That's all I have, please feel free to join the C64 Facebook group if more assistance is needed (there's usually helpful people hanging around there).