Welcome to AA!
Is the hairgrass still in the pot that it came in, with the rockwool? You should remove it from the pot and plant it. It will do better in the substrate. It spreads rapidly and may even spread out of the pot.
How many root tabs are you using? I would use 3-4 at most in a 5 gallon tank.
Sunlight can contribute to algae. I had one tank with pressurized CO2 that was too close to a window and I always battled algae in that tank more than the other tanks.
I had a 5 gallon tank once with sand substrate. I always had string algae in it - large clumps of algae that looked and felt like long hairs. Others have beautiful planted tanks with a sand substrate but I never could figure out how to control that algae in the tank with sand substrate. It lifted out easily but always came back.
What are your water parameters - ammonia, nitrite, nitrate? Do you have a heater? A 25-watt heater is a good size for a 5 gallon tank/bowl. Bettas do best in water that is 80 degrees and doesn't fluctuate much. I have used a sponge filter with a gang valve to control the waterflow in a 5 gallon betta tank. The filter is gentle enough for a betta when controlled by a gang valve. The sponge would give your bacteria more surface area to grow and would probably help to eliminate the bacterial bloom (the cloudy white appearance) sooner than if you just waited for the bacteria to settle on the substrate and other surfaces.
Fertilizers and CO2: Our article section is a good place to get started!
http://www.aquariumadvice.com/articles/categories/Freshwater/Articles/
Here are some "sticky" threads about fertilizing and CO2 - at the top of the Planted Tank forum index page:
http://www.aquariumadvice.com/forums/f24/read-this-first-resources-and-references-83826.html
