That's because the index of $checkbox you want isn't "i", it's $i. I would also recommend concatenating that array piece:
$query = 'DELETE * FROM tbl_name WHERE id=' . $checkbox[i];
Also note that I removed the quotes around the value, since it's (presumably) a numeric value.
You should also be sanitizing user-supplied data with a function such as [man]mysql_real_escape_string/man before you ever use it in a query.