+1 to hiring someone that knows what they're doing if you're looking for professional promotional material. It can get very expensive very quickly if you're looking into video cameras/equipment.
There is plenty of free video editing software that can be used to sync the audio from an external recorder. Avidemux is a simple one available for all platforms.
https://itstillworks.com/12571765/how-t ... n-avidemux
Anymore these days, I think it's quite acceptable to use video off a recent flagship mobile phone with a tripod. Nothing beats familiarity and accessibility when you should be worrying about your performance not setting up camera gear. Lighting, your appearance, and the appearance of the space you're in go a long way towards "video quality". It's relatively simple to get multiple camera angles of the same performance if you know friends or family that also have mobile phones, or if you have something just lying around that you're not using. In terms of bang for the buck, it makes more sense to put your money and time into your audio solution and of course your playing skills, rather than the video equipment.
There exist a myriad of microphone/audio solutions that can connect to your phone, skip all of those. An external recorder with the ability to choose and place your mic, and set your gain appropriately will be a vast improvement in quality.
One of the Zoom recorders like the H4N is a start, as it can be a very versatile tool down the line for many different situations. TASCAM make alternatives that are just as good. The built-in microphone solutions are perfectly acceptable for auditions and even more, as long as you can find a good placement/direction for the space you're recording in. For even better quality down the line, you may want to add an external microphone. If you'll just be recording yourself solo or even with piano accompaniment, a single mic input is good enough. Use relative distance to control the balance of the Piano and Soloist. Which mic/s you end up with is a whole 'nother can of worms. I'd recommend a ribbon mic(be careful with these), anything from the MXL R144 up to the AT4081 should be usable with the R-121 being a good professional standard for this sort of thing.
To anyone that's hiring based on recordings, a great player recorded with a handheld presentation style voice recorder will be preferable to a sloppy player recorded with $10,000 equipment.