Decide what song you want to learn based on what you like best, how well suited the song is to guitar interpretation and your skill level.
Find chords that harmonize correctly with the melody. If you have a trained ear and are schooled in music theory, you can identify the chords yourself. Otherwise, you can find transcriptions of many Anime songs at various chord and lyrics sites on-line. If you want to study the whole song and can read music, sheet music is also available on-line and for purchase.
Use a tablature reference guide (tab chart) to learn the chords you need. Guitar tablature is a simple chart of finger placement on the fret board to create chords. At the beginner level, a basic tab chart gives you the information to play any combination of chords. However, if you are interpreting a song that usually includes a variety of instruments for solo play and if you have some experience, a tab chart that provides options for playing the same chords up the neck may allow you to mimic background melodies more fluidly.
Listen to the song you are learning and strum the chords in time with the lyrics. It may take some time to get the timing of the chord changes just right. Sing along with the music so you can begin to feel the connection between the melody and the accompaniment. Anime songs highlight the lyrics, so singing as you learn the chords will start you on your way to developing an accurate style for your guitar playing.
Immerse yourself in the song your are learning. Anime songs are musically complex yet express defined aspects of the human spirit simply and naturally. Spend some time listening to the whole song many times, identifying all of the rhythms that create the structure of the song.
Use the chord progressions you have learned to play the rhythms you have identified as key to the feeling of the song. Keep practicing until you can play the song to your own satisfaction. Listen, experiment and enjoy.