ADMM steps (from https://web.stanford.edu/~boyd/papers/pdf/admm_distr_stats.pdf) can be
\begin{align*}
{{x}}
&\leftarrow \arg\min_{{{x}} } f \left( {{x}} \right) + {{u}} ^T \left( {{{x}} } - {{z}} \right) + \frac{\rho}{2} \left\| {{{x}} } - {{z}} \right\|_2^2 \\
&\equiv \arg\min_{{{x}} } f \left( {{x}} \right) + \frac{\rho}{2} \left\| {{{x}} } - {{z}} + {u} \right\|_2^2 \\
{{z}}
&\leftarrow \arg\min_{{{z}}} g\left( {{z}} \right) + {{u}} ^T \left( {{{x}} } - {{z}} \right) + \frac{\rho}{2} \left\| {{{y}} } - {{x}} \right\|_2^2 \\
&\equiv \arg\min_{{{{z}}}} g\left( {{z}} \right) + \frac{\rho}{2} \left\| {{{x}} } - {{z}} + {u} \right\|_2^2 \\
{{u}} &\leftarrow {{u}} + \left( {{{x}} } - {{z}} \right)
\end{align*}
Let us say $f(x) = \frac{1}{2} \|A x - b \|_2^2$ and $g(z) = \lambda \| z\|_1$. We can exploit proximal operator, that is,
Definition. Let $f: {\rm dom}_f \mapsto \left(-\infty\right., \left. +\infty \right]$ be a closed convex proper function, then
\begin{align*}
{\rm prox}_{\lambda f}\left( x\right) := \left({I} + \lambda \partial f \right)^{-1} \left( x \right) = \arg\min_{u \in {\rm dom}_f} \left\{ f\left({u}\right) + \frac{1}{ 2\lambda} \left\|x - u \right\|_2^2\right\} .
\end{align*}
Also, define for brevity (we can use the equivalent scaled-form ),
$$F(x) := f \left( {{x}} \right) + \frac{\rho}{2} \left\| {{{x}} } - {{z}} + {u} \right\|_2^2$$
$$G(z) := g \left( {{z}} \right) + \frac{\rho}{2} \left\| {{{x}} } - {{z}} + {u} \right\|_2^2 .$$
Now, just find the gradients and set them to zero, that is,
$$\frac{\partial F(x)}{\partial x} = 0 \Longleftrightarrow \frac{1}{\rho}\partial f(x) + \left(x - z + u \right) = 0 \Longleftrightarrow x = \left(I + \frac{1}{\rho} \partial f \right)^{-1} \left( z - u\right) = \operatorname{prox}_{\frac{1}{\rho} f}\left( z - u\right)$$
and
$$\frac{\partial G(z)}{\partial z} = 0 \Longleftrightarrow z = \left(I + \frac{1}{\rho} \partial g \right)^{-1} \left( x + u\right) = \operatorname{prox}_{\frac{1}{\rho} g}\left( x + u\right).$$
Thus, the ADMM iterative steps are
\begin{align*}
{{x}^{k+1}} &:= \operatorname{prox}_{\frac{1}{\rho}f}\left( z^{k} - u^{k} \right) \\
{{z}^{k+1}} &:= \operatorname{prox}_{\frac{1}{\rho}g}\left( {{x}^{k+1}} + u^{k} \right) \\
{{u}^{k+1}} &:= {{u}^k} + \left( {{x}^{k+1}} - {{z}^{k+1}} \right)
\end{align*}
Now, you can use the prox operators for both affine $f(x)$ and L1 norm $g(z)$.
Appendix
The prox operators for $f(x) = \frac{1}{2} \|A x - b \|_2^2$ and $g(z) = \lambda \| z\|_1$ are given below.
\begin{align}
\operatorname{prox}_{\lambda f}\left( x \right)
&= \arg\min_{v} \left\{ \frac{1}{2} \|A v - b \|_2^2 + \frac{1}{ 2 \lambda} \left\|x - v \right\|_2^2\right\} \\
\Longrightarrow 0&= A^T\left( Av - b \right) + \left(-\frac{1}{ \lambda} \left( x - v \right) \right) \\
\Longleftrightarrow 0&= \left(A^TA + \frac{1}{ \lambda}I \right)v - \left(A^Tb + \frac{1}{ \lambda} x \right)\\
\Longleftrightarrow v&= \operatorname{prox}_{\lambda f}\left( x \right) = \left(A^TA + \frac{1}{ \lambda} I \right)^{-1}\left(A^Tb + \frac{1}{ \lambda} x \right).
\end{align}
\begin{align}
\operatorname{prox}_{\lambda g}\left( z \right)
&= \arg\min_{v} \left\{ \lambda \| v\|_1 + \frac{1}{ 2} \left\|z - v \right\|_2^2\right\} \\
&= \arg\min_{ \left\{v_i\right\}} \left\{ \sum_i \lambda|v_i| + \frac{1}{ 2} \sum_i \left\|z_i - v_i \right\|_2^2\right\}
\end{align}
Since the problem is separable, then you can use KKT conditions to obtain so-called soft thresholding operator. Not to make this post too long, I can refer to you for instance this The Proximal Operator of the $ {L}_{1} $ Norm Function which shows the derivation.
I hope this helps you.
