in this case the guy with the gun has his back turned...
you said exactlly what you wanted to do... so do it.
the function to delete a file is unlink(). use it. that is all the help you need to do exactlly what you want.
your current code:
// accept file from user
// add file to file system and update db
new code:
// accept file from user
unlink("path/to/old/resume.txt");
// add file to file system and update db
or:
// accept file from user
if( // add file to file system and update db ) {
unlink("path/to/old/resume.txt");
}