Playing accompaniment on the violin is very different from playing a melody - the focus is on playing chords in a rhythm that backs up the other instruments, playing "second fiddle" in a sense, to fill out the sound and texture of a song.
Choose your chord. The chord you choose depends on the key of the song you are accompanying. G and D chords are good choices, as they are common keys for many songs, and because one of the notes can be played on an open string.
Place your fingers to play all the notes of the chord you've chosen. If you have chosen a G chord, yplay the notes G, D, and B, which can be done with G and D on the lowest two open strings and B with your first finger on the A string.
Angle your bow and take a few practice strokes to make sure that you can hit at two of the three notes at once. With the G chord, the lower two notes contain the root of the chord, or the note it is named after; practice playing first the lower two, and the the upper two notes of the chord. Playing accompaniment is all about rhythm, so the playing style needs to be more staccato than melodic.
Choose your rhythm. The rhythm should fit both the meter and the style of the song. The most basic pattern is to play one stroke on each beat of the measure. For a song that is in 4/4 time, start out playing one short stroke on each beat in a low-high-low-high pattern. For a song that is in three or six, a low-high-high pattern captures each beat of the measure and gives the rhythm a nice swing.
Vary the rhythm once you are comfortable with the basics. Keeping the same notes, try playing them shorter and off the beat, or just every other beat, or two per beat. These variations help your accompaniment stay interesting and fit the style of the song.
Claire Blaustein is a freelance writer and music critic whose work has appeared in a variety of publications, including NPR.org, The Washington Post, Exclaim! Magazine, BBC Music Magazine, and La Scena Musicale. She has a master's degree from McMaster University in Music Criticism, and runs several websites - Ear to Ear Project, a music blog, and The Barefoot Kitchen, a food blog.