As horrible as this looks, it's not bad. Only selecting the visible vertices gives you one way to control the vertices on which you are working. It's a tool, but it's no good unless you have control over it:
Press Ctrl + Z to undo deleting Suzanne's vertices.
Press 1 on the NumPad to select the Front View.
Now, look at the 3D View header. There is a button that controls the visibility of the hidden and back-facing vertices, edges, and faces. It is highlighted on the right-hand side of the following illustration. It is just to the right of the three buttons that control whether you are working on vertices, edges, or faces. Left-click on it so that it turns light grey. You'll notice that you can see more edges within Suzanne.
Press A to deselect all the vertices.
Use the Border Select tool to select all of the vertices.
Press X. A menu pops up. Select Vertices.