Without the DLC, the only place you can find wild Froakie is the single individual at the top of the tower in Wild Zone 16. Thus, your best option for shiny hunting it is to do the fast travel method where you repeatedly reopen your map and fly to Wild Zone 16 over and over. This Froakie has a 5% chance of being Alpha with no way of increasing your odds, so you have about a 1/81920 chance (~1/20487.6 with Shiny Charm) of getting your desired shiny Alpha. This will likely be a rather difficult hunt, especially since it's harder to automate fast travel with external tools.
If you come across any non-Alpha shinies that you want to catch, make sure you save, since Froakie is a skittish Pokémon who will run away forever if it sees you while you're trying to catch it. Fortunately, any Alphas will attack you instead of fleeing.
With the DLC, your odds can be significantly improved, as members of the Froakie evolutionary line can be found in 2-star, 3-star, 4-star, and 5-star Water-type distortions. You can reroll your distortions by sitting at a bench.
The best possible donut to use is a Sweet donut with Lv. 3 Sparkling Power and Lv. 3 Alpha Power. This gives you a 51% chance for each spawn to be Alpha, resulting in a ~1/2007.84 chance (~1/1148.18 with the Shiny Charm) of getting a shiny Alpha with each spawn, and there will generally be groups of multiple spawns. Using 8 Tanga Berries is the best way to try to get this donut, although it is random. More info here on how to reset until you get it. When you're in the distortion, you can repeatedly fast travel to the entry portal to reroll spawns, though you'll need to approach the group as well if they're far away from the portal.
Lv. 3 Sparkling Power also guarantees a shiny upon entry. If you use Sparkling Power that's Water-type or All Types, it might affect a different Pokémon instead of one from the Froakie line. However, if you find a 4-star or 5-star distortion that contains Greninja, you can use a Dark-type Sparkling Power to guarantee a shiny Greninja. It still only has a 51% of being Alpha, but you can reroll whether or not it's Alpha by using the backup save trick I linked to.
