Is there such a formula? Given a set of data for which the mean, variance, skewness and kurtosis is known, or can be measured, is there a single formula which can be used to calculate the probability density of a value assumed to come from the aforementioned data?
There are many such formulas. The first successful attempt at solving precisely this problem was made by Karl Pearson in 1895, eventually leading to the system of Pearson distributions. This family can be parameterized by the mean, variance, skewness, and kurtosis. It includes, as familiar special cases, Normal, Student-t, Chi-square, Inverse Gamma, and F distributions. Kendall & Stuart Vol 1 give details and examples.
This sounds like a 'moment-matching' approach to fitting a distribution to data. It is generally regarded as not a great idea (the title of John Cook's blog post is 'a statistical dead end').
D’Agostino’s K2 test will tell you whether a sample distribution came from a normal distribution based on the sample's skewness and kurtosis.
If you want to do a test assuming a non-normal distribution (perhaps with high skewness or kurtosis), you'll need to figure out what the distribution is. You can look at the skew normal distribution and the generalized normal distribution. If you do this, you consider other distributions too.