How do I grep through binary files that look like text?

You can use grep anyway to search through the file – it does not really care if the input file is really text or not. From ‘man grep’:

    -a, --text
          Process a binary file as if it were text; this is equivalent to the --binary-files=text option.

   --binary-files=TYPE
          If  the  first few bytes of a file indicate that the file contains binary data, assume that the file is
          of type TYPE.  By default, TYPE is binary, and grep normally outputs either a one-line  message  saying
          that a binary file matches, or no message if there is no match.  If TYPE is without-match, grep assumes
          that a binary file does not match; this is equivalent  to  the  -I  option.   If  TYPE  is  text,  grep
          processes  a  binary  file  as  if  it  were  text; this is equivalent to the -a option.  Warning: grep
          --binary-files=text might output binary garbage, which can have nasty side effects if the output  is  a
          terminal and if the terminal driver interprets some of it as commands.

Please mark the words of caution at the end of the second paragraph. You might want to redirect the results from grep into a new file and examine this with vi / less.