That is an excellent question. It rather depends on each individual design.
In order for a cross stitch chart to work on needlepoint canvas, you want to find a design that does not use fractional stitches, as they are difficult to work in Canvaswork stitches.
Needlepoint can be worked all in one stitch, such as tent stitch, or in a variety of stitches.
You would work one tent stitch (or half cross) in place of each cross stitch on your chart. The fibre used would depend on the size of the canvas you are using. For example, with an 18 count needlepoint canvas you might like to stick with stranded cotton floss, but for a 10 count canvas you might choose a tapestry wool instead.
Simple designs, with blocks of color and not too much shading, can be worked in a variety of stitches to give added texture. For example, you could work a Santa's beard in french knots or velvet stitch.
I tend to stick with smaller stitches for detailed areas, leaving the bolder effects for background areas.
One thing to keep in mind is that often you won't be able to complete an entire stitch as you reach the outer edges of an area. Here you would implement what is known as compensating stitches, where only part of the stitch is worked.
You can get really creative and have lots of fun experimenting. I would love to see your results.