Like any other artistic domain, singing lends itself perfectly to self-learning. You can learn to listen to your own voice and correct notes that are out of tune, adjust the vocal cords and vocal timbre, master breathing and, little by little, you can start calling yourself a singer. It's a lot like yoga or self-disciplined sports. You always have to develop patience, inner mental strength, breathing technique through listening to yourself, feeling yourself, and a good vocalist, because you have to approach all that even with a mentor.
I highly recommend Berklee Vocal for Performance with Donna McElroy DVD, and then some practice, and then some silence, and then some listening, and rinsing and repeating. If you want to learn more about your mixed voice and how to find it, this video is a good place to start. Too many people just try to raise their voice from their chest as high as they can when they are learning to put on their belts. So if you want to improve your respiratory support, increase vocal power and learn to sing in tune, learn good respiratory support.
To reach your full potential as a vocalist, you'll need to be able to switch between the head voice and the chest voice, using something known as a mixed voice.
Singing lessonswill help you considerably, but if you can't take them, there are still ways to learn on your own. A simple way to train the ear (and also learn the vocal range) is to play a note on a piano and then try to match the pitch you hear. If you're self-taught, by definition, you don't have much experience, and therefore you won't be able to do things like listen to a vocal fault and diagnose the problem (if you realize a problem in the first place), let alone understand how to correct that problem without causing a new problem.
The chest voice refers to playing the notes in the lower part of your vocal range; learning to project these notes creates a vibration that you can feel in your chest. If you mean learning how the voice physically works and understanding the anatomy, physiology and physics behind it, or learning music theory skills with books, then yes, that is also possible, although it is probably quite difficult, time-consuming and possibly expensive if you want to buy textbooks and other materials. An important step in the development of singing is learning to sing with what is called the “lead voice”. However, if you're like most people who participate in vocal training, you probably want to be able to hit warm high and low notes.
If you mean learning to consistently produce beautiful and healthy sounds with your voice, while minimizing unhealthy or harmful habits, and learning to stylistically apply them to the repertoire that is appropriate for your voice and development. What this means is that you need to learn to mix the voice of your chest and the voice of your head to create a “mix”. You probably already know if you play a drastically wrong note, but it's impossible to measure your exact vocal tone by listening to you sing. To minimize the risk of injury and keep your voice as healthy as possible, be sure to practice some vocal warm-up exercises before each practice session.
The good news about learning to sing from home is that you can start with some basic and fundamental techniques to master that have nothing to do with singing (but everything, as you will soon learn).