Find the version of the song that is appropriate for your playing level. If you are new to the violin, play a simpler version found in a hymnal or online at music stores like J.W. Pepper or Virtual Sheet Music.
Look for more difficult music if you are an intermediate player. The music in the hymnal may be used, but play using double-stops, rather than just playing the top notes of the chords, as you would if you were a beginner.
Play more advanced versions of "Amazing Grace" either of your own invention or from hymn books written especially for the violin. These may be purchased at your local music store or online at J.W. Pepper, among other stores.
Practice a scale in the key of the musical arrangement. Then play the music slowly. If you are playing a beginner's rendition, then you may not need to practice the music a great deal before performing it. Play in first position throughout, being certain to use vibrato if you are comfortable doing so.
Go through a scale in the same key as the music you are playing if you are intermediate. Play a version of "Amazing Grace" with double-stops. In practicing these "mini-chords," it is necessary to slow down and listen to the bottom note as you play. When it is in tune it is easier for the finger playing the upper note to be in tune. You may want to play the bottom note first, then the top note, then both together to get used to the finger positions each double-stop requires.
Play a scale in the key or keys of the version you will be using if you have an advanced version of the hymn. Key changes and variations using chords, high octaves, high finger positions and sixteenth-note passages may be some of the elements in the piece. Practice each of these sections individually, if necessary, before practicing the entire piece all the way through.