How to build a self-made Qi compatible (wireless) charger
The Qi wireless charging protocol involves a complicated communication handshake from receiver (target device) to transmitter in order to enable charging. It is not trivial and requires strict timing. You 100% could make your own coil and driving circuit but it would be an endeavor for the sake of bragging to do it. The standard is open but you need to be a member of the wireless power consortium to get full access. See http://www.low-powerdesign.com/article_TI-Qi.html for some details.