I've never used the tile cleaner before. What's in it? As long as it doesn't bubble or foam up when scrubbing you're probably OK.
Our tile is also in need of cleaning and I think I've talked the wife into doing it today. My Alk & PH are on the low sides (90/7.2) so I plan on using a solution of baking soda & water. What I've done in the past is just float around the perimeter scrubbing while the baking soda & water bucket is on the edge. It can make a mess on the concrete coping however it is easily rinsed off. The trick we discovered last year is to rinse the tile thoroughly by sloshing pool water on it