Assuming you've made above the Federal and State amount for filing a self-employment form (Form C), you would fill out the Form C (self-employment reporting), which asks about expenses related to your business.
You can file as a "sole proprietor" of your business. The form does ask for a simple explanation of the nature of the business.
For an expense as substantial as a computer, you should keep purchase receipts, credit card statements, etc. in case you are asked to include/provide them.