Looking for characters within a string

You can test each item in the first string for being in the second string; at any “non match” stop and return false.

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\NewDocumentCommand{\checkTF}{mmmm}
 {
  \stringchecker_if_string_in:nnTF { #1 } { #2 } { #3 } { #4 }
 }

\bool_new:N \l__stringchecker_temp_bool

\prg_new_protected_conditional:Nnn \stringchecker_if_string_in:nn { T, F, TF }
 {
  % set the temporary boolean to true; it will remain such
  % unless a nonmatch will happen
  \bool_set_true:N \l__stringchecker_temp_bool
  % map over the substring to find
  \str_map_inline:nn { #1 }
   {
    % do nothing if there is a match; ##1 is the current item, #2 the string to search in
    \str_if_in:nnF { #2 } { ##1 }
     {
      % a nonmatch breaks the mapping and sets the boolean to false
      \str_map_break:n { \bool_set_false:N \l__stringchecker_temp_bool }
     }
   }
  \bool_if:NTF \l__stringchecker_temp_bool
   {% there was no nonmatch, return true
    \prg_return_true:
   }
   {% a nonmatch was found, return false
    \prg_return_false:
   }
 }
\ExplSyntaxOff

\begin{document}

\checkTF{asdf}{asdfgh}{TRUE}{FALSE}

\checkTF{asdf}{hgfdsa}{TRUE}{FALSE}

\checkTF{asdf}{asdghj}{TRUE}{FALSE}

\checkTF{}{asdghj}{TRUE}{FALSE}

\checkTF{a}{}{TRUE}{FALSE}

\end{document}

A slightly more efficient version that absorbs one token at a time from the first argument.

\documentclass{article}
\usepackage{xparse}

\ExplSyntaxOn

\NewDocumentCommand{\checkTF}{mmmm}
 {
  \stringchecker_if_string_in:nnTF { #1 } { #2 } { #3 } { #4 }
 }

\prg_new_protected_conditional:Nnn \stringchecker_if_string_in:nn { T, F, TF }
 {
  \__stringchecker_check:nN { #2 } #1 \q_nil
 }

\cs_new_protected:Nn \__stringchecker_check:nN
 {
  \quark_if_nil:NTF #2
   {
    \prg_return_true:
   }
   {
    \str_if_in:nnTF { #1 } { #2 }
     {
      \__stringchecker_check:nN { #1 }
     }
     {
      \__stringchecker_stop:w
     }
   }
 }
\cs_new:Npn \__stringchecker_stop:w #1 \q_nil
 {
  \prg_return_false:
 }
\ExplSyntaxOff

\begin{document}

\checkTF{asdf}{asdfgh}{TRUE}{FALSE}

\checkTF{asdf}{hgfdsa}{TRUE}{FALSE}

\checkTF{asdf}{asdghj}{TRUE}{FALSE}

\checkTF{}{asdghj}{TRUE}{FALSE}

\checkTF{a}{}{TRUE}{FALSE}

\end{document}

\documentclass{article}

\usepackage{xinttools}

\makeatletter
\newcommand\mycheck[2]{%
   \xintFor*##1in{#1}:
   {%
      \in@{##1}{#2}%
      \unless\ifin@\expandafter\xintBreakFor\fi
   }%
   \ifin@\expandafter\@firstoftwo\else\expandafter\@secondoftwo\fi
}
\makeatother

\newcommand\test[2]{#1 in #2 ? \mycheck{#1}{#2}{TRUE}{FALSE}}
\begin{document}

\test{asdf}{asdfgh}

\test{asdf}{hgfdsa}

\test{asdf}{asdghj}
\end{document}