The immediate answer to your question is, Google for "delaware incorporate", and look at the ads. I think we used Agents and Corporations, Inc., who were the cheapest at the time.
IIRC, to incorporate in Delaware, you pay ~$100 for a filing fee. Since you don't live in Delaware, you're also required to have a Registered Agent in the state, which also costs money and is built into the fee structure for whichever company incorporates you. If you just pick the cheapest turnkey incorporation service off Google, you're probably not going to beat the cost trying to DIY.
To answer some questions you didn't ask:
Are you sure you want a C corp? C corps can be liable for corporate taxes. If you're a 2-person shop, C corp status probably isn't what you want; you don't care about having a board of directors, and you want pass-through taxation. What you want instead is an LLC, which requires less paperwork but still provides you with limited liability, a tax ID, and an entity that can invoice and be invoiced.
Are you sure you want Delaware? You probably do, if only to save time and energy thinking about it. There's a NV vs. DE debate with alleged tax advantages for NV, but if you're concerned about startup costs, DE is probably the cheapest and most hassle-free. For instance, we looked in to incorporating in NY or IL (our home states), but both required us to run an ad in the paper announcing ourselves!