The above creates a hatch with diagonal lines from the lower left to the upper right that are 4 pixels apart. Besides the diagonal line (M0,4 l4,-4) you also have to stroke the upper left and the lower right edges of the pattern area, since the line will otherwise be "constricted" due to clipping where it intersects the edges of the square.
One problem with drawing a diagonal line within a pattern is that when the pattern is tiled the lines won't always line up - especially at high zooms. (It depends on the SVG rendering engine you happen to be using).
@Ingo's answer above attempts to resolve this by drawing in the triangles at the top-left and bottom-right corners - but again, using some rendering engines and high zooms, it doesn't always look best - and sometimes the line ends up looking a bit like a string of sausages.
Another approach is to draw a horizontal line in the pattern and rotate the pattern, e.g.
Use the patternTransform attribute to rotate a vertical (or horizontal) line segment. This method tiles seamlessly and uses the simplest possible path. The pattern width attribute controls how close parallel hatches are.
The background path is overly-wide on purpose, so that there's no part of the pattern not at least covered by the background. Meanwhile, the hatches can have their width tuned to change how thick the lines are.
Some great points got lost in the comments, so I wanted to aggregate that knowledge with some simpler inline examples. As far as I can tell, what method you choose to use is entirely up to preference since the heavy lifting is being done by patternTransform + rotate either way, but personally I think the <rect> method is easier to digest at-a-glance even if it might make more semantic sense to use <line>.
SVG Non-Scaling Pattern with <rect>
Define the spacing between your lines with the <pattern>'s width, and the width of the lines themselves via the nested <rect>'s width.
While the above examples solve the issue, SVG 2's Hatches (Candidate Recommendation 2016) are a documented method of tackling this exact issue. Inkscape happens to implement them because one of its developers, Tavmjong Bah, was an Invited Expert at the time, but browsers are still tackling them. To note, the latest Editor's Draft (2018) doesn't include hatch, which doesn't necessarily mean anything, but might be why it hasn't yet been prioritized by browser vendors.
Tracking SVG 2 Features for the Browser
Each major engine has an ongoing thread for feature implementations: