This involves some trial & error and in the end, it's debatable which result you might consider to be the "best result". So allow me to just give some generic advice:
-flatten option to get rid of transparent background. The transparency makes it hard to judge actual quality of the result. If you need the transparency in the final image, you can remove
-flatten once you're sure of the quality.
use something like
-density 300 to get a high DPI result. The main issue with
convert is that it uses a very low density by default (72 DPI). This parameter has to be specified before the input file.
downscaling a high DPI image might cause additional blur, so perhaps calculating the correct DPI value to achieve the desired resolution is the way to go:
$ convert -density 100 file.pdf -flatten file100.png
$ file file100.png
file100.png: PNG image data, 827 x 1169, 8-bit colormap, non-interlaced
$ echo $((1080*10000/1169))
$ convert -density 92.38 file.pdf -flatten file9238.png
$ file file9238.png
file9238.png: PNG image data, 764 x 1080, 8-bit colormap, non-interlaced
I'm not sure if there is a way to have
convert determine "ideal" DPI value by itself.
If you take this question to the ImageMagick IRC channel or forum, I'm sure you'd get some more advice. It helps if you provide the link to the PDF file you're working with. ;)
You can also improve quality in other ways, for example by trimming empty borders away. You're losing a lot of resolution if half of the page is white. There are even solutions that re-wrap PDF text to get the most out of available screenspace (e.g.
Finally, also try other programs. This is a matter of opinion, but I prefer using Inkscape or GhostScript directly. ImageMagick has characters "glued together", Inkscape has a more balanced result, and GhostScript allows you to render a blur-free pure pixel image (if that's something you like - use
pngalpha for the blurry version, which is virtually identical to
gs -r92.38 -sDEVICE=png48 -sOutputFile=ghostscript.png file.pdf