Start with a Normal probability plot https://en.wikipedia.org/wiki/Normal_probability_plot . The tails are where non-Normaility can be most extreme. If the probability calculations are going to make on the distribution are roughly in the "middle" of the distribution, as opposed to the tails, you can get away with a lot. But if you're trying to do risk analysis, say, and need to understand the probability (risk) of an extreme outcome, the tails of the distribution (and one of those tails in particular) is all that matters. Of course, dependence vs. independence across random variables or events can also be a huge driving factor in what the actual probability of an extreme event is.
So, along the same lines as whuber's comment, you have to base your assessment of closeness to Normality on why you care how close the distribution is to being Normally distributed and what you're going to do with the distribution.