Grungy Bass Tone?

Sounds like someone needs a Rickebacker!

Seriously, by "grunge" what do you mean by comparison?

choose a crunchy guitar rig that has the kind of distortion you are looking for and enable the Parallel Path feature.

now you can shape the DI of the bass with stomps A and B, the distorted part of the sound with the rest of the blocks and set the mix of the two sources.

What Don said. I do it in my DAW, since there's a little more flexibility - one track for the DI, heavy compression, EQed to give you a steady low-end and some nice "clank", and another track reamped through whatever the guitars on that song are running, EQed so only around 200Hz-1000Hz is coming through.

What this tells me is that I really have to play more with Parallel Path more.

Quote from Lokasenna

What Don said. I do it in my DAW, since there's a little more flexibility - one track for the DI, heavy compression, EQed to give you a steady low-end and some nice "clank", and another track reamped through whatever the guitars on that song are running, EQed so only around 200Hz-1000Hz is coming through.

right.
I also like to compress the DI quite a bit, then add bass and carve out a good chunk of the low mids/mids (or 'wool').
make the distorted part of the sound mid-heavy/reduce bass (Green Scream).

That is the standard way of doing it.

Either looking for more clarity with a Fender or more gurgle for a Rick, you need 2 distinct tracks, one to do the work of the low end, the other in the mid range. But playing has much a big part (the orchestration part), since the mid-range will interfere with so much in the mix, you only want a busy bass when other things, notably singer, hang back to let it come through, or else it starts getting in the way. Unless getting in the ways is what you intend...

I just asked about your idea of "grunge" cause it's not well defined, and even "grunge" bands chaffed at the designation, and have such bass timbre variety, it's hard to know your specific interest.