You say you have problems with the whites, that sounds like your screen is set too bright, most people's are. If your screen is much brighter then the average of your field of view, you're essentially staring at a light source, that causes strain for the eyes and facial muscles. And of course makes the tones hard to evaluate. This means you may need to adjust the brightness over the day, if your computer is in a room with daylight. A good starting point is that white sections of the screen, like the standard background of this website, should be about the same brightness as a piece of paper held next to the screen.
To take the guesswork out of what is white and black, you should get familiar with histograms and tone curves. Google those, maybe also together with the software you use. The histogram for the picture you posted has a peak a little away from the left margin, and is empty to the left of that. That means there is no real black in it. Once you understand the histogram, the tone curve tool is the most intuitive way to change it, and in the simplest terms, that means changing contrast. But whereas with the "contrast" slider it is a stab in the dark where which tones come out, if you use the tone curves tool, you know exactly what is happening. The levels tool is an alternative some prefer.
It also helps to have a bit of black and a bit of white on the screen as a reference, many photo editors allow for that.
Your choice of words sounds like you think of "post processing" as " enhancing", tempering with the picture, and are uncomfortable with that. Don't. This is exactly what you do in the darkroom when you pick a paper grade and exposure time - you're matching the contrast from the capture to the viewing medium.