First off, as a point of clarification, your aim should not be to increase pH. Your aim here should be to build alkalinity, which will in turn normalize and stabilize pH, along with alkalinity and calcium.
I've never heard of lime juice (as in the juice squeezed from limes, right?) being used for this purpose.... the chemical used is calcium hydroxide, otherwise known in the hobby as kalkwasser (which is a german term for limewater, which may be the source of confustion??). If lime juice is actually used, I'm interested to hear about it though...
Generally, it's used by mixing about 1tsp per gallon of the powder into RODI water. The water is then dripped very slowly into the tank. Adding it too quickly can cause a tank crash, so you want to be sure to drip very slowly at first, then increase as necessary over time to get the alkalinity, calcium, and pH values you are looking for. All three of these values play off each other, so all should be monitored.
You can buy kalkwasser mix from Seachem, Kent, etc at most lfs stores that sell saltwater stuff. Alternatively, you can buy calcium hydroxide powder from a site like bulk reef supply, or you can get Mrs. Wages Pickling Lime from the grocery store, it's the same stuff.