I recently upgraded myself from a small 30 watt amp and a Fender P-Bass knock-off...

...to a Crate BT100 and a Squier Frank Bello Jazz Bass. I'm starting to get better tones already with this, but I want to achieve a distinctive metal bass sound without the use of a pick.

An example of a bass tone I want is in the song The Light That Blinds by Shadows Fall. Can anyone help me with this? What should I do on my amp? Do I need to buy pedals? Do I need new strings?
well, firstly youd probbly get more/better advice in the bass forum. an amp like a peavy, or hartke would give a more metal tone, an od/distortion would help aswell. you could try the boss 0db-3, or an ehx big muff (or wait till the bass version is released). oh, nice choice of bass.
It will be hard to get that tone with that equipment. Generally Jazz basses don't give you a "metal sound". To get a distinct sound finger style, you want to boost the low-mids to cut through. Too much will sound muddy - especially with a Jazz bass.