The Erdős-Kac theorem gives that, for a fixed function $M(N)$ with $\limsup M(N)/\log \log N < 1$,

$\frac{1}{N} |\{n\in[N,2N] : \omega(n) > M(N)\}|\rightarrow 1$.

Likewise if $\liminf M(N)/\log \log N > 1$,

$\frac{1}{N} |\{n\in[N,2N] : \omega(n) > M(N)\}| = o(1)$,

where here $\omega(n)$ counts the number of prime divisors of $n$ without multiplicity. (So $\omega(4) = \omega(2) = 1$, while $\omega(6) = 2$.) The same results will be true if prime factors are counted with multiplicity however. (i.e. we consider $\Omega(n)$, where $\Omega(4) = 2$ for instance.)

More precise asymptotics can be obtained, especially easily for $M(N) = o(\log \log N)$, by using formula of Sathe and Selberg, and its extensions. These are uniform versions of the theorem of Landau which has been mentioned by quid. Where $M(N)$ grows like $\log \log N$ or faster, I'm afraid these formula become somewhat complicated, and I wouldn't expect a nice asymptotic expression (but I could be wrong). A reference is "On the number of prime factors of an integer" by Hildebrand and Tenenbaum, the easiest offshoot of which (due to Sathe) is that Landau's formula holds uniformly for $k = o(\log \log x)$. Formula (1.7) of Pomerance will give you (with a little patience) nice upper bounds.

The book of Tenenbaum already mentioned is also nice reference for some of these questions, as is chapter 7 of Montgomery and Vaughan's "Multiplicative Number Theory I". Kac's book is great for anyone to read, interested in these questions or not.