An approach to determine whether language is regular or not?
I'm not aware of any completely "algorithmic" method to decide whether a language is regular.

For the specific language you are interested in, you can start by observing that it can be written as combination of unions and intersections of a finite number of regular languages:
$$
\bigcap_{\alpha \in \Sigma} \left( L_{2,\alpha} \cup L_{3,\alpha} \right),
$$
where:


$L_{2,\alpha}$ is the language containing all words $w$ such that $|w|_\alpha$ is divisible by 2.
$L_{3,\alpha}$ is the language containing all words $w$ such that $|w|_\alpha$ is divisible by 3.


It is easy to show that all these languages are regular (since, in each language, $\alpha$ is now fixed). A DFA for $L_{2,\alpha}$ has two states $q_0, q_1$ where $q_i$ means that the number of characters equal to $\alpha$ encountered so far is $i$ modulo 2. A DFS for $L_{3,\alpha}$ is similar but has $3$ states.

The regularity of $L$ follows from the fact that regular languages are closed under union and intersection.