I'm almost certain that you currently have something like this:
move_uploaded_file( $_FILES[ 'video_field' ][ 'tmp_name' ] , $destinationFolder . $_FILES[ 'video_field' ][ 'name' ]
You should put the API code after that line and then replace
'file.mov'
with
$destinationFolder . $_FILES[ 'video_field' ][ 'name' ]
.
E.g.
$filesource = $yt->newMediaFileSource($destinationFolder . $_FILES[ 'video_field' ][ 'name' ]);
$filesource->setContentType('video/quicktime');
$filesource->setSlug($destinationFolder . $_FILES[ 'video_field' ][ 'name' ]);
Simple.