your query is correct, to avoid duplicates, after inserting a first table, get a last inserted id, and stored that id in other tables,
$query = ('INSERT INTO name (fname,lname,..) VALUES ('Mary','Blige',..); ') // here, set custid as autoincrement
$res = mysql_query($query)or die(mysql_error());
$custid = mysql_insert_id ($res);
$query1 = ('INSERT INTO address (custid,town,parish,..) VALUES (1,'Mona','t. Andrew',..); ')
$query2 = ('INSERT INTO phone (custid,cell,fax,..) VALUES (1,8760001234,8769230002,..); ')
$query3 = ('INSERT INTO computer (custid,brand,type,..) VALUES (1,'Dell','Laptop',..); ')
mysql_query($query1)or die(mysql_error());
mysql_query($query2)or die(mysql_error());
mysql_query($query3)or die(mysql_error());