If the chain is jumping when you dial up the power, usually it's an issue of a worn chain, worn cassette, or worn chainrings up front, or a combination of all three. Looking at your photo I can tell this bike has had a lot of use, needs cleaned, and was not exactly top of the line to begin with. My guess is that you need to update/replace chain and cassette at least. Also a thorough cleaning, degrease, and relube is probably in order, and while you're at it the cables almost certainly need some attention if the rest of the bike is in this condition.

Yes, you can easily do this yourself, but you will need some specialty tools. Specifically a chain breaker tool and at least a cassette lockring tool. The parts will run you $30-$50 depending on where you get them and what quality you buy. Then there's the knowing how, which is another thing altogether.

Also, there are some compatibility considerations, so you will need to know exactly what you are looking for, and it would probably benefit you to have the assistance of a shop or a friend who is knowledgeable about bikes. Good luck!