Since you ask for formulas for the character, I will first assume that you are interested in finite dimensional representations.

If $\mathfrak{g}$ is a basic classical Lie superalgebra, such as $\mathfrak{sl}(m|n)$, one distinguishes between typical weights and atypical weights. A weight is typical if the irreducible representation $L(\lambda)$ is projective, i.e. does not have any nontrivial extensions. For atypical weights one can further distinguish between the degree of atypicality $1,2, \ldots$. For typical weights Kac already gave a character formula in

- Kac, V. G.: Characters of typical representations of classical Lie superalgebras.

For atypical weights no closed formula is known in general. For weights which have atypicality 1 there is a character formula obtained by Kac and Wakimoto in

- Kac, Victor G.; Wakimoto, Minoru: Integrable highest weight modules over affine superalgebras and number theory.

This covers in particular the case when $\mathfrak{g}$ is a simple exceptional Lie superalgebra or $\mathfrak{osp}(2|2n)$ or $\mathfrak{sl}(n|1)$ since in these cases dominant integral weights are either typical or have atypicality 1. This leaves basically the cases where $L(\lambda)$ is an irreducible representation of atypicality $\geq 2$ of $\mathfrak{gl}(m|n)$, $\mathfrak{osp}(m|2n)$, $\mathfrak{p}(n)$ and $\mathfrak{q}(n)$ for general $m,n$.

In the $\mathfrak{gl}(m|n)$ case the character problem was succesfully first solved by Serganova in

- Serganova, Vera: Kazhdan-Lusztig polynomials and character formula for the Lie superalgebra $\mathfrak{gl}(m|n)$

However this does not give you a closed formula for the character. It is basically an algorithmic solution. Her approach is very similar to the usual one in category $\mathcal{O}$: Write down an (infinite) resolution

$$ 0 \leftarrow M^0 \leftarrow M^1 \leftarrow \ldots $$

of $L(\lambda)$ which has a filtration with quotients isomorphic to Kac modules (the universal highest weight modules in these representation categories). The character of $L(\lambda)$ is then

$$ ch L(\lambda) = \sum_{i=0}^{\infty} (-1)^i ch M^i $$

where the characters of the $M^i$ can be easily calculated since the characters of Kac modules are known (by Kac himself). Then finding the character of $L(\lambda)$ amounts to determine the coefficients $b_{\lambda,\mu}$ in

$$ ch L(\lambda) = \sum_{\mu} b_{\lambda, \mu} ch V(\mu) $$

where $V(\mu)$ is the Kac module. The coefficient $b_{\lambda,\mu} = K_{\lambda,\mu}(-1)$ is the value at $-1$ of a certain Kazhdan-Lusztig polynomial. A nice overview article of Serganova's work is Gruson's Bourbaki article

- Gruson, Caroline: Sur les representations de dimension finie de la super algebre de Lie $\mathfrak{gl}(m,n)$.

Later Brundan gave a different approach using categorification techniques in

- Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebra $\mathfrak{gl}(m|n)$

A closed formula for the character was then obtained in

- Su, Yucai; Zhang, R. B.: Character and dimension formulae for general linear superalgebra.

They actually obtained a closed formula for the character by reworking some calculations of Brundan. The formula however is so complicated (with an immense amount of cancellations) that this is not comparable to the nice situation of the Weyl character formula for a semisimple Lie algebra. In some cases (when the weight is a so-called Kostant weight) the combinatorics collapses and one can get simple formulas. This can be understood in terms of KL theory since Kostant weights are precisely those, in which the occurring KL polynomials are monomials.

Algorithmic or Kazhdan-Lusztig type solutions for the character problem are known also for other Lie superalgebras such as $\mathfrak{osp}(m|2n)$ (due to Gruson-Serganova and later by Cheng-Lam-Wang) and $\mathfrak{q}(n)$ (due to Penkov-Serganova and Brundan), but in general no closed formulas are known for general weights. As far as I am aware the problem of finding the character in the periplectic $\mathfrak{p}(n)$-case is open so far.

Of course the character problem has been studied for irreducible modules in category $\mathcal{O}$ as well. As others have mentioned, relevant names here are Brundan, Cheng, Lam, Wang and many others. For the infinite case I would recommend the overview article by Brundan

- Brundan, John: Representations of the general linear Lie superalgebra in the BGG category $\mathcal O$.