I go camping to France every year. There is a wifi station at 200 metres from where I camp, but I don't have a direct line of sight (it's on a slope). I can put a shoebox at the base station, but I prefer not to. I am willing to write the software myself. There is no cell reception and I don't have a HAM licence. How can I transmit data from and to the station?
I'm not familiar with the regulations in France, but if they are similar to the US, then an amateur radio license would not help you, because the traffic of a typical internet connection (HTTPS encrypted data, most notably) would be illegal to transmit.
Assuming that's the case, and considering your question purely as a matter of using unlicensed WiFi gear:
Putting equipment at the base station would not help you (much), since it's under the same restrictions as the existing equipment. Instead, what you would have to do is find an accessible location which has line of sight to both you and the base station. There you would place a WiFi repeater (also known as an extender) and a battery sufficient to power it.
The repeater will need to be configured with the SSID of the target network. It must be a model which works with arbitrary networks; many will only work with access points of the same brand. (Or at least, that was the case some years ago; I haven't researched the current state of WiFi much.)
You will need a battery with sufficient capacity to power the repeater over the time it is operating. If the repeater location is inaccessible (so that you're not turning it off when you're not using it) then this may be a serious problem. Check the specifications of the equipment for its current draw before buying, and test it before your camping trip.
You will also need to supply the repeater with the proper voltage. This can be done by using a battery with the proper voltage (possibly composed of several cells/batteries wired in series), or by using a DC-DC converter (power adapter). The converter adds inefficiency but also adds regulation, allowing proper operation over a much larger voltage range and therefore use more of the capacity of the battery.
Most likely you will want to use a single 12 volt battery and a matching adapter, as that is a common voltage due to its use in cars. However, you should not use a car battery as they can spill acid and are not designed for complete discharge, in addition to being very heavy. A better choice would be a lithium iron phosphate battery.
You will also need a battery charger designed for the type of battery.