The first thing I'd recommend is to define the bass tone you're after.
Once you've determined that, it's just a matter of having the right gear.
ie: Are you looking for a straight DI tone... or an amped up tone?
For straight DI, the Avalon U5 and A-Designs REDDI are both popular choices... as is the Tech-21 VT Bass.
If you're after a DI that includes AmpSim, I'd recommend the Zoom B3.
The B3 is one of the least expensive easiest solutions to recording Bass.
The B3 can output both processed signal (AmpSim) and straight DI simultaneously.
At $199, there's no better solution.
If you're into classic-rock bass tones, your Fender J running thru the B3's Bassman model will sound great.
If you have the recording space, a nice bass head and cab, and the mics to do it well... mic'ing a real bass cab sounds fantastic. I like to use a pair of mics on a 2x10 cab. An SM57 captures a tight-focused mid (great for the attack and articulation)... and either a LDC (condenser) or LDD (dynamic) captures a more "full range" tone.
Mix the two mics to taste...
If you're recording DI... and you're found your preferred DI box, it's super easy to record with consistent results.
You can read books and watch videos for ideas...
But (IMO) there's no substitute for diving in and learning by doing.
Create a simple backing track... and start trying various options.
Recording via straight DI, the Bass will have extended frequency response.
Deeper lows and higher highs.
As a player, you'll have to work harder to control dynamics... as there's no amp to smooth things out.
Pre and/or Post compression can help.
Recording a bass amp, the speaker will filter off the highs and deepest sub-bass.
The amp will smooth out dynamics.
The amp will tend to "toughen up" the sound (when compared to straight DI).
FWIW, I love Fender basses.
I tend to not like a J straight DI (with no AmpSim).
To my ears, a J recorded straight DI sounds a bit "anemic".
Even if you've got a killer J and a world-class DI...
Play that J thru an Amp or a quality AmpSim... and it's fantastic.