The simple rule: If content can reach a person's eyes or ears, it can't be protected. The more hurdles you put in place to protect content, the less users will want to use your product.
So, assuming david-ing-name's answer isn't an option, there will almost certainly be no way to keep someone from capturing the content.
My attempt, for a client running on a user's general purpose computer to which they have complete control, i would use images to disable cut&paste. In other words, all messages are presented as an image to the receiving end. You can even go so far as to make it a captcha-style image so that OCR will be difficult. To deal with screen-shots, you can only put hurdles in place but one idea, sticking with the image captchas, is to require users to enter a captcha to be able to read content, otherwise the text is simply unavailable to the client. For the user to continue to read the content, they must keep the mouse over a small area of the screen and/or hold down a certain key combination. Specifically, something that would prevent them from changing focus to another application. Still, with all that in place, anyone reasonably skilled could get around all this fairly easily by, for example, simply cracking the client. And there is no technique available to handle cracking the client unless you're writing a webapp. But with a webapp, the screen-capture becomes easier to crack and harder (impossible?) to implement.