I've used a mixture of vinegar and baking soda mixed with water, and also lemon juice with baking soda and water, and let it sit filled up with that each mixture about an hour. I covered the bowl with a one of those big black plastic garbage bags to make sure no bird tried to use it. Then scrub with a nylon brush, have even used metal brushes but be careful you don't overdo that, you could remove some of the finish.
Then I coated the inside with a sealer and that made it so much easier to keep clean. Before the sealer I'd have to scrub as much as the initial cleaning and never get it really clean unless I used vinegar solutions again. With the sealer on it, I just swish a nylon brush around, dump and rinse and fill it back up. The algae still grows but it doesn't stick like on the untreated concrete. I used
Ghostshield Siloxa-Tek 8510 for the sealer according to their instructions and I love it. It didn't change the appearance at all but made it so much easier to clean. I ordered just the small 16 oz size, was just under $20 including taxes and if any shipping. I'm going to recoat 2 bowls this spring with it, it lasts a long time.
There's no way to eliminate the algae, it will keep growing back. There are some drops you can add (
Fountec) and a teaspoon of apple cider vinegar per gallon, but it still won't eliminate algae, you will still need to keep after it.
Edit: I wouldn't recommend using anything like CLR. Not only might it pit the concrete but it might be absorbed into the concrete and leech out in the water which might poison the birds. I wouldn't worry about vinegar and lime juice as much.