Not expert on Tensorflow, and I heard it is quite complicated to learn, and that's why they are making the 2.0 version which seems to be in RC right now.

Many start learning neural network by training a simple XOR function. I think the XOR function teaches some fundamental principles of machine learning, the "gotchas", and basic training methods. You can definitely build a simple NN code yourself with backpropagation.

For minmax games, you are applying the same concept but the input to your neural network is your board configuration and the output is how much it "likes" it, typically within the range of [0, 1.0] based on your activation function.  Then you apply the same principle in minmax trees and pick the path it favors the most based on the tree's depth.

The training for minmax games however is different than backpropagation. All machine learning requires a learning path, to know where to go. It is easy to train XOR function using backpropagation because 1 XOR 1 = 0, and we can tune the network to get as close to 0 as possible. Minmax games, not as easy, because what's the best move? Who gets to decide it is the best move?

If you have a robust preexisting AI as a benchmark that you can use to train your NN with, feel free to use it. Otherwise, you can also train with evolution. I have trained a minmax game using evolution against a random CPU (a CPU that just picks a random move), and whichever NN with the most points scored over N number of games get to "breed" with random mutation. It yields some decent results, enough to beat me.

Google offers a crash-course.
It uses videos and text (mainly video lectures) but with 20hours it should fit nicely with your 1 week time span.