One of the most instantly recognizable looks for punk rockers is the mohawk. Although it might be more acceptable to sport a mohawk these days, the hair style still is mostly associated with punk. If you feel up to it, you can go for a full mohawk, which involves shaving the sides of your head and leaving a strip on top that is spiked up with hair product. For a simpler style, just trim the hair on the sides of your head, and spike up the middle.
Dyeing your hair an unusual color, such as pink or blue, also can fit well into a punk costume. You can use a kit to actually dye your hair, which can take a while, or use a spray-on hair-coloring product, which will not look as good but will be gone fairly quickly.
A simple way to get a punk look is to take an old T-shirt and cut it up so there are several rents in the garment. Use a razor blade, and wash the shirt afterward so the rips will look more worn, rather than deliberate.
Putting patches on your clothing can be good way to make a punk costume. Sew a big patch onto the back of your jacket or hoodie and one or two onto your pants. Preferably, these patches will have the name of a punk band on them, or some kind of radical saying, but anything can work.
Bright red and black plaid pants or skirts can make a good part of the lower part of your punk costume. For pants, the tighter the better, and "bondage pants" (pants with lots of zippers and clasps) are specifically associated with the punk look. Also, black leather pants and jackets were worn by some punk bands.
Piercings are a good accessory for a punk costume. Fake piercings on your lips, eyebrows and nose can add a lot. If you already have piercings, try putting a safety pin (after disinfecting it) into the piercing hole for your costume.