Examine the work itself. The signature should look like the signatures the artist has signed on other works. The style of the work itself should match the artist's style of painting. For instance, you would expect a painting by Seurat to use pointillism, while Van Gogh is known for for vivid colors and heavy brushstrokes.
Examine the labels on the back of the painting. Museums and galleries affix labels to the back of paintings before they hang them for display. If one of the labels is from a gallery that was known to have sold the artist's works, you can contact the gallery to authenticate the painting. If the gallery is out of business, the label will help if it is from a gallery that was known to have worked with the artist. Museum labels often contain information about the artist, and you can contact the museum listed on the label to confirm that the painting in question was by a particular artist. Auction labels often simply list a lot number with no other information, but if you have access to old auction catalogs, you may be able to match the lot number to a listing in a catalog.
Take the painting to someone who is an expert on the artist. The best choice is the author of the artist's catalogue raisonné -- the authoritative list of the artist's known works, compiled by an expert. The expert will have exhaustively studied the artist's works and, upon examination, will give his pronouncement on whether the painting was done by the artist.
Send the painting for forensic testing. Consider whether the potential sale price of the piece is worth this expense. In 2009, a forensic investigator found a fingerprint on a painting, then matched the print to another on a known Leonardo da Vinci painting, which raised its value from about $20,000 to $160 million. Forensic labs also carbon date the canvas, frame or paint itself and examine the brushstrokes at a microscopic level to make sure they match the brushstrokes of known works by the artist.