How to set replacement text of function from token list variable?

The main problem is the parameter text that must not be braced. You can skip it by using the “without parameter” syntax. See https://tex.stackexchange.com/search?q=cs_set%3ANV for examples of usage

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\tl_set:Nn \l_tmpa_tl { \textbf{#1} }

\exp_args:NNV \cs_new:Nn \foo:n \l_tmpa_tl
\cs_set_eq:NN \foo \foo:n

\cs_show:N \foo

The output on the console is

> \foo=\long macro:#1->\textbf {#1}.

You can also do with the parameters, but it's really ugly:

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\exp_args_generate:n { NNNV }

\tl_set:Nn \l_tmpa_tl { \textbf{#1} }
\tl_set:Nn \l_tmpb_tl { #1 }

\exp_args:NNNNV
\exp_last_unbraced:NNV
\cs_new:Npn \foo \l_tmpb_tl \l_tmpa_tl

\cs_show:N \foo

There is the p argument type, which is a wrapper around TeX's #{ argument. The advantage is that anything valid in the parameter text of a macro is valid here without changing the code. The downside is that the argument that follows must be braced, so you'd either need to change the expansion type from V to o (which will work because of an implementation detail), or to pass a braced, single-token argument to V (which will work because of another implementation detail).

To stick with the expl3 syntax and have the braces required for p you can use v:

\documentclass{article}
\usepackage{expl3}
\ExplSyntaxOn
\exp_args_generate:n { Npv } % p = #{
%
\tl_set:Nn \l_tmpa_tl { \textbf{#1} }
\exp_args:NNpv \cs_set:Npn \foo #1 { l_tmpa_tl }
% \foo=\long macro:#1->\textbf {#1}.
\exp_args:NNpv \cs_set:Npn \bold text #1 { l_tmpa_tl }
% \bold=\long macro:text#1->\textbf {#1}.
\ExplSyntaxOff
\begin{document}
\foo{hello} \bold text{world}
\end{document}