I could write a book on this topic, but it would not compare to e books already done by Alan Murphy:
https://www.alanmurphyphotography.com/shop
https://www.alanmurphyphotography.com/product-page/the-guide-to-songbird-setup-photography
In addition to the postings above,
I use a 400mm or a 500mm lens;
Distance from the lens to a single perch placed by me is very close to the minimum focus distance of the lens.
Lens aperture is set to wide open; Sometimes it is closed down a bit to allow or more DOF on the subjectt;
Perch placement is set so background is as clean as possible and free of light reflections.
If background is hard to control, raise or lower the perch and/or your shooting position so a cleaner and more uniform background is present, like green grass;
If water is not available use a 5 gallon water jug and have it drip onto a surface so the drip creates noise.
Adding color to the background can be done by adding potted plants that are blooming; hand made colored plywood, etc.
Background rendering is impacted by how close you are to the subject; how far the subject is from the background; and the aperture of the lens used. In my backyard setups, this is the order I follow to control backgrounds, assuming you do not change the focal length of the lens. Generally speaking a longer focal length lens works better than a shorter one (for background control) but MFD of each lens plays a role too.