The number of positive integers that are products of a given number of primes follows an asymptotic similar to the prime number theorem. These numbers are called $k$-almost primes. 

More precisely, the counting function $\pi_k(x)$ of numbers that are the product of $k$ primes is asymptotically: 

$$\frac{1}{(k-1)!} \frac{x}{\log x } (\log \log x)^{k-1} $$


In particular, the density behaves like $\frac{(\log \log x)^{k-1}}{\log x }$   and thus is $0$. 
For a lot  more details see https://mathoverflow.net/questions/35927/asymptotic-density-of-k-almost-primes