Actually I have been trying to implement the same where I have two numeric columns x and y and I want to plot those points using different color and size. I went ahead to plot line chart with points for them , but it didn't seem to differentiate the points of x and y. Meanwhile data looks like this.
Do you have a column that codes the groups in your pca_coordinates? I guess not.
Maybe you need to merge the results from the PCA with your input.
When you have this column, you need to define this in the aes() statement,
ggplot(pca_coordinates, aes(x=PC1, y=PC2, colour = group))
See comment from scottyd22 to show us how your data looks like.