I am not sure I get your understanding of peakedness and heaviness. Kurtosis means "Excess" in German, so it describes the "head" or "peak" of a distribution, describing whether it is very wide or very narrow. Wikipedia states that the "peakedness" is actually described by the "kurtosis", whereas peakedness does not to appear to be a real word and you should use the term "Kurtosis". 

So I think you might have gotten everything right, the head is the Kurtosis, The "heaviness" of the tail might be the Skewness":

Here is how you find it: 

$$
a_3 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^3}{N * s^3_x}
$$

with s as the standard deviation for x.


The values indicate:

Negative Skew:
$$
a_3 < 0
$$

Positive Skew:
$$
a_3 > 0
$$

No Skew
$$
a_3 = 0
$$

You can get a value for the kurtosis with:
$$
a_4 = \frac{\Sigma^{N}_{i=1}(x_i - \overline x)^4}{N * s^4_x}
$$


The values indicate:

Platycurtic: 
$$
a_4 < 3
$$

Leptocurtic:
$$
a_4 > 3
$$

Normal:
$$
a_4 = 3.0
$$

Did that help?