Making a multi-track video can be a challenge. There are a wide array of processes, software and tools which you have to choose from and going from camera to schmick final product can be a daunting prospect. To that end here are some tips and tricks to help streamline the process:
Prepare a guide track. This is essential as it means your quartet will stay in time with each other. Often its easiest to let the lead or bass (or member with best rhythm) lay down their video first for the rest to sing to!
Record your video! A phone will generally be pretty great for this. Just remember to have plenty of light - move those lamps around until your face is clear and bright and don't be afraid to use props in the background
Use software to combine the videos. This can be as complicated as you're able. If you have some technical skill iMovie and Final Cut Pro are great tools for preparing a multitrack video. Otherwise A Capella (mixcord) and WeVideo are free app and web based options which are readily available. Downside is A Capella is only for iDevices and costs a small subscription to have videos longer than 30 seconds, and WeVideo leaves a watermark on the end product
Export the video and share! Most of the free and paid software options include ways to share the video directly to Facebook or Youtube, or to save it to your PC/Phone. Then you can do what you will!
If your sound is too echoey or sounds like a wet paper bag (boxy) you can liven or dampen the sound by adding more sound sponge surfaces. For example if the room has an echo, placing rugs, sheets etc over hard surfaces will dampen the sound. If the issue is outside noise, use pillows or a mattress over the door and windows to stop that pesky noise from coming in!
If you're having trouble singing your part on its own, try singing along to a past recording of your quartet singing the song. Or ask the lead to sing the melody and you can sing along to that!
Practice. This one might sound obvious to some, but singing and performing on video has its own difficulties which need practice to really nail! Practice taking your video, watch it back, take notes, and repeat!