You cannot. You can measure moments of the distribution (average, standard deviation, skewness, kurtosis, and so on) but unless you have all the moments exactly (an additional problem) you cannot determine the distribution, at best you can get an approximation.
I agree with dear Gert: It is impossible to determine a unique density function from data. But you could postulate many possible density functions from the data and also to contrast or to test its fit to a determinated density function.