First thing: don't write to a file you're reading from, you're likely to end up erasing it. Use the inplace-edit feature (-i) instead.
Next, inside hard quotes ('), variable substitution doesn't operate, so you'll need to do something else to get the expanded $PATH substituted in there.
Lastly, $PATH is very likely to contain / characters, which would break the substitution syntax for sed, so you'll need to use another separator.
Assuming your paths never contain ;s, try (after having backed-up your current file of course):
sed -i 's;^PATH=.*;PATH='"$PATH"';' .bashrc
The match part of the substitution means: match a string that starts with (^) the string PATH= and followed by any character (.) any number of times (*). I.e. it will match lines that start with PATH= completely, but not lines that have PATH= somewhere in the middle. The .* is important because you want to replace the whole line (try without it to see what happens).
The quoting to get $PATH substituted inside the replacement is necessary to account for cases where $PATH would contain whitespace characters.
Demo:
$ foo=bar
$ echo hello | sed -e 's/hello/$foo/'
$foo
Env. vars are not substituted inside hard quotes. So you get a literal $foo there. Now let the shell expand it by bringing it out of the hard quotes:
$ foo=bar
$ echo hello | sed -e 's/hello/'$foo'/'
bar
Ok! But this isn't safe:
$ foo="bar baz"
$ echo hello | sed -e 's/hello/'$foo'/'
sed: -e expression #1, char 11: unterminated `s' command
sed received two arguments after -e above: s/hello/bar and baz/, separately. Not good. To fix that, you need a bit more quotes:
$ foo="bar baz"
$ echo hello | sed -e 's/hello/'"$foo"'/'
bar baz
Putting an environment variable inside " quotes is something you should pretty much always do when you're passing it as an argument to something else, otherwise (assuming $IFS hasn't been changed) it will get split on whitespace.
