Hm, that’s a good question! Pretty much any modern book on, or with a chapter on, multilinear algebra, or even a text with a modern treatment of differential geometry (smooth manifolds) would, I imagine, do a decent job. Also, searching the web for “tensors as multilinear forms” turns up quite a few promising links. For a video lecture that could give you a flavor of what tensors are from the linear-algebraic perspective, try
