Rovin, sorry to say it, but on an HDTV without adjusting the settings, the graphics are going to look the same and, in some cases, worse. If your cartridge is grainy, then chances are you have a problem with your system and/or game, unfortunately. My cartridges (yes, I still play them) have crystal clarity, though they are blocky and slightly pixelated, there is absolutely no grainy-ness.
As far as your issue, Ultimatium, here's what I have to suggest.
Here's the deal, it's an HDTV and, unlike most situations, that's unfortunate. That is because most of these lack interlacing/scanlines, which are what make the sharp, jagged edges appear on the HDTV. All CRT monitors/T.V.s (what the original game was made for; the VC is a direct emulation of the cartridge) have the scanlines. Scanlines were implimented intentionally back in the day with old game systems to hide the jagged edges that the hardware wasn't able to then easily clear up with anti-aliasing rather than waste precious ROM space to make it better, since it was more expensive to put that into a system like the N64.
Your best bet is to check your T.V.'s settings and see if it offers "interlacing" or "scanlines." If not then the next best thing would be to turn down the sharpness and/or (on some TVs) resonance (I believe it should be called). This will lower the overall quality, but blend the lines a little to cover up the spaces that used to be overridden by the scanlines. Resulting in slightly less clear pictures, but objects that exist in three dimensions much more realistically. As if you were looking through slightly blurred vision (on a blocky, low-graphics world, obviously).
Hope that explaination makes sense, and good luck with your graphics issue!