Obviously at least 2 colors are necessary. Color all points $(x, y)$ with even sum of coordinates with one color and all other points with another color. All points that are at distance 5 from $(x, y)$ are $(x \pm 4, y \pm 3)$, $(x \pm 3, y \pm 4)$, $(x \pm 5, y)$, $(x, y \pm 5)$. In each case, the sum of coordinates has the parity different from that of $(x, y)$. Therefore they are colored differently from $(x, y)$.