Rendering a triangular face in 3d space

Let's say I have a triangular face in 3d space, and I have the 3d coordinates of each vertex of this triangle, and would also have other information about the triangle(angles, lengths of sides, etc.). In Java, if I have the viewing screen and its information, how can I draw that plane, without using libraries like LWJGL, to that image, assuming I can properly project, accounting for perspective, any 3d point to that 2d image.

Would the best course of action just be to run a loop that draws each point on the plain to a point on the image(i.e. setting the corresponding pixel), which will most likely set the same pixel multiple times? If I'd do this, what would be the best way to identify each point in an oblique triangle, or a triangle that doesn't line up nicely with the axes?

tl;dr: I have a triangular face in 3d space, a "camera" looking at the face, and an image in which I can set each pixel. Using no GL libraries, what's the best way to project and draw that face onto the image?