Here is the fingerspelling chart. Practice the letters, both making them and reading them. Reading is harder. You will almost always be able to spell faster than you can read.
A great way practice is to sit down for 5 minutes and start spelling every word you see. Do that a couple of times a day and your speed will increase drastically. Focus on keeping the flow and tempo smooth. Also be sure not to bounce the letters. That makes it very hard for someone to read. Don't try for speed. Fluency is the key, speed comes later!
Double letters are done one of two ways. Closed handed letters, like A, S, M, etc and doubled by opening the hand and reforming the letter. You do not have to open the hand all the way, just relax back to the neutral position. Open handed letters like L and B are done by maintaining the hand shape and sliding the letter to the right (or left if you are spelling with your left hand).
Remember to practice spelling with both hands.
Try spelling words to another person and vice versa. Reading is much harder to do and requires more practice. When trying to read someone's fingerspelling it is important to approach it in a similar manor as reading the words in a book. If we are reading the word car, we don't read C-A-R. Instead we read ka-ah-rr. Fingerspelling is similar.
There is a really neat fingerspelling tool here. This is great to use in the beginning because you can see things fingered the right way and you can adjust the speed.
If your hands start to get sore, stop. Fingerspelling is very hard on the hands. Far more so that signing words.
For an extra challegne, try learning the song for next time.