Let $\mathbb{N} = \{1,2,\dots\}$ be the set of natural numbers, and let $C_0$ the the collection of functions on $\mathbb{N}$ that vanishes at infinity. Let ${\cal C}\subset C_0$ be the collection of functions that is finite linear combinations of $f_s(n) = s^n$ where $0\le s<1$. More explicitly, $$ {\cal C} = \left\{\sum_{j = 1}^k a_if_{s_i},\, k\ge 1,\, s_i \in [0,1),\,a_i \in \mathbb{R}\right\}. $$ Here, it is closed under multiplications and is a linear subspace, so it is an algebra, it separates points and vanishes nowhere. Hence by Stone-Weierstrass it is dense in $C_0$ in uniform norm.
Now, I want to get an explicit sequence of functions in ${\cal C}$ so that it converges to the indicator function at n = 2, call it $1_{2}$, that is, $$ 1_2(n) = \begin{cases}1& \text{if n = 2}\\ 0&\text{else}.\end{cases} $$ Or if it is easier, find an explicit sequence of functions in ${\cal C}$ so that it converges to $1_{1,2}$, that is $$ 1_{1,2}(n) = \begin{cases}1& \text{if } n \in \{1, 2\}\\ 0&\text{else}.\end{cases} $$ Since the function $\frac{1}{s} f_s(n)$ converges uniformly to $1_1(n)$ as $s\downarrow 0$.
More generally, can one find an explicit sequence to approximate indicator function at $k \ge 1$ for arbitrary $k$?
For $j=1$, as you stated, we have $$s^{-1}f_s\to 1_1.$$
For $j=2$, we get $$s^{-2}f_s-s^{-3}f_s\to 1_2.$$ This is because $$s^{-2}f_s=\Bigl(s^{-1},1,s,s^2,\ldots\Bigr),$$ and $$s^{-3}f_{s^2}=\Bigl(s^{-1},s,s^3,s^5,\ldots\Bigr).$$ The difference is $$s^{-2}f_s-s^{-3}f_{s^2}=\Bigl(0,1-s,s-s^3,s^2-s^5,\ldots\Bigr).$$ Taking $s\downarrow 0$ gives convergence to $1_2$.