There are a few way to turn a chest of drawers into a bookshelf cabinet. You can use the chest cabinet as a ready-made structure and just add shelves, or you can turn the drawers on their sides to get to a modular, modern look as wall-mounted singular shelf units.
Using the Chest Cabinet
Remove the drawers and measure the width and depth of the cabinet from the inside. Cut your wood to those measurements. Cut as many pieces as you had drawers. Alternately, if the drawers have flat bottoms and don't have dove joints or other complicated cuts, you can dissemble the drawers and use the bottoms as shelves.
Take those pieces of woods and secure them into the cabinet where the drawers once slid. There's already a little ledge there where the draw rolled in, so it should be easy to position these shelves and secure them.
Paint the inside of the bookshelf cabinet and the shelves, both on top and bottom. Sand and repaint the outside if you wish. If you like, cut more wood and mount on the bottom of each shelf to disguise the old shelf ledge. Keep in mind that if the cabinet isn't tall, this really won't show.
To keep it from looking like a chest that you've converted into a bookshelf, you can use thin molding or even tiny strips of finishing wood like luan plywood to finish the front edges of your cabinet, both on the sides and in front of shelves. Attach these strips first and then paint for the best look.
For a modular look, remove the drawers from the bookshelf and mount them to your wall. As the drawers should be straight and angular, this will eliminate your need to measure and cut wood.
Remove the drawers and remove any hardware on them, like pulls and rollers. If there are holes that will show in the finished project, you can fill them in with wood putty. Paint or stain the drawers however you want. A good look is to paint the insides a different color than the outside, especially if you paint the bottoms of the drawers (the back of the "shelf") a different color than the rest.
Finishing Drawer Bookcases
You can mount individual drawers on the wall as small floating modular bookshelves. Alternatively, you can bolt them together on their sides to make a taller bookshelf and set it on the floor or mount it to the wall.
You can make these as fancy or as plain as you want. Wooden embellishments and moldings are available in a variety of shapes. If you have one converted drawer mounted on the wall with the sides showing, you can add paint, stripes or molding here to create a very unique and interesting piece.