Each country has its own sign language, and each sign language has its own alphabet. Like spoken words, fingerspelling each letter of each word is cumbersome and tedious-- y-o-u w-o-u-l-d-n-'t w-a-n-t- t-o s-p-e-l-l o-u-t e-v-e-r-y w-o-r-d y-o-u s-a-y o-r h-e-a-r (or read). However, knowing the alphabet is helpful for names, locations, and as the basis for many words. Knowing the alphabet is valuable when learning ASL because many signs incorporate a letter in their formation.
For the most part, communication is much better when signing actual words, along with contextual clues. However, there are times when knowing the alphabet is helpful in communicating. For example, proper names and locations are spelled out the first time they mentioned, followed by pointing to a spot to establish a reference location for the person or place. In further references, they are usually identified by a namesign (if there is one) or by using the first letter and pointing to the reference location.
When learning a biblical story, signing even the first letter of a name or location can help with muscle memory. For example, if I am trying to remember the name Joab in telling a story from 2 Samuel or 1 Kings, I might sign "J" while I say the name aloud. If I forget the name, I may still remember the sign, and the name becomes a bit easier to remember.
To tell a story, I may use the first letter of a name or place to establish a physical location. As Jesus and his disciples are going to Jerusalem, I may again sign "J" and point ahead after the first reference to the city. Jesus would always be signed with the traditional sign (fingers to mark nail holes in palms -- see post on basic signs.) This could be confusing for an audience if I am telling a story that includes the same beginning letter for multiple places or people (like Jerusalem and Joab in 2 Samuel 11,12), but that doesn't happen too often. To distinguish a location from a person, I may add a sign for city and have reference locations in different places.
In the video below, I am signing the alphabet. Generally, hands do not move around when signing a letter, but to give a full view, I will sign the letter, rotate my hand, and then sign it again in the original position.