What is shape of data. 

Shape of data is determined by how we choose to look at data\visualize data. How do we look at data is dependent on our choice of various dimensions of the data. So  basically our choice of dimensions describes shape of data.

Covariance Matrix describes shape of data.

Diagonal elements of covariance matrix capture variance of data along different dimensions. Non diagonal values of covariance matrix determines the covariance between different dimensions of data.
So the shape of data is dependent on our choice of dimensions and is captured pretty well by covariance matrix. 

What is one ideal way of looking at data. 

Ideal way of looking at data is when we choose dimensions so that covariance between those dimensions is minimized to zero. And does that mean for covariance matrix. It means that we try to diagnolize the covariance matrix. In an ideal world we can perfectly diagnolize the covariance matrix. So that the new set of dimensions now have ZERO covariance between them. 

PCA is essentially trying to diagonalize the covariance matrix by trying to find orthonormal eigen vectors of covariance matrix