How would I simulate a network to explore the percolation threshold of a network connected by the knight's move?
First of all, you don't simulate an infinite board. You simulate larger and larger boards, until the percolation threshold seems to stabilize. For a given size of board, you need to decide on what event signifies that percolation happens. One common option is that the top of the board is connected to the bottom of the board. For each probability $p$, you estimate the probability that this happens by doing many samplings. You then use binary search to estimate the probability $p_c$.

Here are some more details. Suppose you've decided on some board size. The first step is to compute all the edges, that is all pairs of vertices connected by a knight's move. Given a probability $p$, you run the following experiment many times. Put in each edge with probability $p$ independently. Then check (using DFS/BFS or equivalent) whether the top of the board is connected to the bottom of the board (that is, add a new "top" vertex connected to all vertices at the top of the board, add a similar "bottom" vertex, and check whether the two are connected). Do this many times, and estimate the probability $\theta(p)$ that "bottom" is connected to "top". Then use binary search on $p$ to find a value of $p$ such that $\theta(p) \approx 1/2$, say. This is your estimate for the critical probability.