Let $F$ be a nonarchimedean local field, and $C_c^{\infty}(F^{\ast})$ the vector space of locally constant complex valued functions of compact support. For $\chi$ a character of $\mathcal O_F^{\ast}$, extend $\chi$ to a function $\chi_{\ast}$ on all of $F^{\ast}$ by setting $\chi_{\ast}(x) = 0$ for $x$ outside $\mathcal O_F^{\ast}$.
I have read that $C_c^{\infty}(F^{\ast})$ is spanned by the functions of the form $\chi_{\ast}(ax)$ for $\chi$ a character of $\mathcal O_F^{\ast}$ and $a \in F^{\ast}$, and was wondering why this is.
The translations $\chi_{\ast}(ax)$ give you functions on $\{x \in F^{\ast} : \nu(x) = -\nu(a) \}$.
If $U_n = 1 + \mathfrak p^n$ is contained in the kernel of $\chi$, then if $x_1, ... , x_t$ is a set of representatives of $U_n$ in $\mathcal O_F^{\ast}$, then
$$\chi = \sum\limits_{i=1}^n \chi(x_i) \textrm{Char}(x_iU_n)$$
I need to somehow isolate the characteristic functions of the cosets $x_iU_n$.
As mentioned above, you need to isolate the characteristic functions of the cosets of $1+\mathfrak p^n$ in $\mathcal O_F^{\ast}$. Let $m = [\mathcal O_F^{\ast} : 1 + \mathfrak p^n]$, and let $U_1, ... , U_m$ be the cosets of $1+\mathfrak p^n$ in $\mathcal O_F^{\ast}$. There are $m$ distinct characters of $\mathcal O_F^{\ast}$ which are trivial on $1+\mathfrak p^n$, and each of these is a sum of the form
$$c_1 \textrm{Char}(U_1) + \cdots + c_m \textrm{Char}(U_m)$$
Characters of an abelian group constitute a linearly independent set, and so they form a basis for the $m$-dimensional vector space $\mathbb{C} \textrm{Char}(U_1) + \cdots + \mathbb{C} \textrm{Char}(U_m)$.