A thermal airship (sort of like a hot air balloon with a propeller) is likely the cheapest way to build an airship. It requires no helium and the lifting gas is just heated air. 

 

I made the envelope out of polyurethane-coated ripstop nylon and sewed it into a cylinder shape. I made a small version first as a prototype. It seemed to work well. I made the burner with a liquid propane tank (basically a forklift tank). It connects to heating coils by the burner flame that vaporizes the propane before sending it to the output. This liquid propane setup is capable of generating a massive amount of heat, enough to rapidly heat enough air to generate lift, unlike a regular propane tank (trust me I tried). The small version is not capable of lifting a person. However, I will be building a larger one soon.