You don't need an expensive toolkit to build your own computer, but there are a few simple tools that you absolutely must have. All of these can be easily obtained at almost any computer store or online, usually in a little case that contains everything you need. You can also buy most of them at ordinary hardware stores.

As with all tools, you should buy the best quality computer tools that you can afford. Good tools last longer and make the job easier. All computer tools should be non-magnetized.

You will need, at a minimum, the following simple tools:

You should have available, at a minimum, small- and medium-sized Phillips and flat screwdrivers and a 1/4" nut driver.

Very small needle-nosed pliers are very handy for removing and inserting jumpers on motherboards and hard drives. (A pair of tweezers or surgical forceps also works well for this purpose.)

Plastic cable ties are useful for neatly bundling wires and cables away from fans and other components inside the computer. If you can't find them, electrical tape is an acceptable substitute.
 

This is a little elastic strap connected to a wire with an alligator clip on the other end. The alligator clip is attached to a metal part of the computer chassis, and the elastic band slipped around your wrist. Better-quality anti-static kits also include a rubberized mat that connects to the wire and is placed under the computer being worked on. This provides extra static protection, and also protects your tabletop from scratches. 

This is usually included with processor fans, but may be purchased separately. It is applied neatly to the area where the processor contacts the heat sink to improve cooling efficiency. (Some heat sinks have the compound "built-in" behind a little peel-off label.) I like Arctic Silver.

This is really needed only if some of your parts are used and/or dusty. Never blow into a computer with your lips to remove dust. Your breath contains too much moisture (and maybe other things depending on what you ate for dinner). Use canned air instead. It's cheap.

You'll need a pill bottle or other small container to hold the various screws, jumpers, and other small parts used to assemble and configure a homebuilt computer.