Chain gets worn when it is not lubricated, or when it is dirty (even if lubricated at the time).
To thoroughly clean the chain, it needs to be taken off the bicycle, cleaned in some solvent that cleans dirt and oil (or whatever lubricant was used on the chain). Mineral spirits do a good job. Put it in a jar, shake it, rinse in water, repeat, until clean. Old mineral spirits can be filtered and reused for the purpouse.
Then it needs to be dried, then lubricated.
Quick way to somewhat clean the chain is to run pedals backwards, and hold a rag over the chain. This method leaves dirt between the rollers and pins, but does get some dirt off the outer side of the chain.
Lubricating chain - one drop of lubricant onto each roller, then spin for about a minute to let the lubricant work it's way between the rollers and pins, then use a cloth to hold the chain and clean the extra lubricant off the outer part of the chain - lubricant is needed only between rollers and pins, on the outside it only gathers dirt.
Chain needs to be cleaned and lubricated when it starts making noise, usually. Though some dry lubes leave a chain noisy, even while it is lubricated (teflon for one).
I like cheaper chains so I don't waste much time and mess my flat while cleaning and lubing. Just use the worst method - rag over the chain while spinning, then add lube it with motor oil and that's it. Some clean chains thoroughly. Each to their own.
Dry lubes get less dirty and are good for dry environments. For wet environments, oils are better, but they get more dirt as well. Dry lubes need a very clean chain before they are applied. Oily ones are less demanding.
Wrote a blog post explaining the basics:
Best bicycle chain lube