An $(\infty,2)$-category is the special case of $(\infty,n)$-category for $n=2$.
It is best known now through a geometric definition of higher category.
Models include:
the definition by Carlos Simpson and Tamsamani;
the definition in terms of n-fold Segal spaces;
a definition in terms of scaled simplicial sets, following Verity’s simplicial model for weak omega-categories by Jacob Lurie (see reference below)
See also the list of all definitions of higher categories at (∞,n)-category.
In (∞,2)-Categories and the Goodwillie Calculus Jacob Lurie discusses a variety of model category structures, all Quillen equivalent, that all model the (∞,2)-category of $(\infty,1)$-categories, in generalization of the standard model category models for (∞,1)-categories themselves (see there for details).
Recall that
A simplicially enriched model category with respect to the standard model structure on simplicial sets hence models ∞Grpd-enriched categories, hence (∞,1)-categories.
Along this pattern $(\infty,2)$-categories should be modeled by categories enriched in the Joyal model structure that models the (∞,1)-category of (∞,1)-categories.
Write $SSet^J$ for SSet equipped with the Joyal model structure. Then, indeed, there is a diagram of Quillen equivalences of model category structures
between Joyal-$SSet$-enriched categories, Joyal-$SSet$-enriched complete Segal spaces and simplicial Joyal-simplicial sets.
This is remark 0.0.4, page 5 of the article. There are many more models. See there for more.
Classes of examples include
For $\mathcal{C}$ a suitable monoidal (∞,1)-category there is the $(\infty,2)$-category $Mod(\mathcal{C})$ of $\infty$-algebras and $\infty$-bimodules in $\mathcal{C}$. See at bimodule - Properties - The (∞,2)-category of ∞-algebras and ∞-bimodules.
(∞,2)-category