I have $N$ random values and I initially know that it is not a Normal distribution (it is a discrete one), but it is really close to that. I estimate the expectation and variance using my number set and approximate the probability distribution by Normal one with empirical parameters. How can I estimate the quality of the approximation (error rates)?
There are a number of distances between probability distributions in the literature. One that you can use here is the Kolmogorov distance (also referred to as Kolmogorov-Smirnov statistic), which is the maximum difference between the two cumulative distribution functions. The Hellinger distance should also work and probably some others (some of which are more difficult to compute). See https://en.wikipedia.org/wiki/Statistical_distance under "Examples".
Note that you are currently in your histogram presenting the normal distribution as if it were discrete, but in fact it isn't. The Kolmogorov distance can be computed between one discrete and one continuous distributions, the Hellinger distance (like a number of other distances) requires both to be continuous or both to be discrete; you can use this only after having "converted" the continuous normal into a discrete distribution, which you apparently have done.
I don't quite know what exactly you want to do with your "estimate" of quality (which a distance measure can be taken to be), so I can't comment on which exact distance to choose. The question whether the normal approximation here is "good enough" for whatever you want to do is somewhat different from giving a precise measure, and I agree with the commenters that this looks good enough for pretty much any standard thing formally requiring normals.
