Instead of using the painting tool to assign blend weights for rigging, how can I manually key in the weight for each vertex?
In the right (N) panel in the 3D view, there's a section called "Vertex Weights". It lists all the weights the active vertex is assigned to, and you can change any of them here.
Note that this only works for the active vertex, not all the ones you have selected. If you want to assign the same weight to a group for all the selected vertices, go to the Mesh tab in the Properties panel, go to the Vertex Groups section, find your group and set your weight, and hit Assign.