I see lots of people using the Focusrite Scarlett 2i2 interface - around $140-$160. Gives you 2 inputs for guitars/mics/inst using either TRS or XLR connection.
An excellent and extremely affordable DAW (software) is Reaper. You can download and use the full version freely - you can purchase it for $60. Purchase allows for frequent updates.
I second both suggestions. I have the Scarlett 2i4, it has a couple extra tricks over the 2i2 that would be very useful if this interface is all you have between your amp and the computer (mainly better direct monitoring control). They didn't bring this design forward into the 3rd-gen line, but 2nd-gen versions are still available. The 3rd-gen replacement is the 4i4, which adds two extra line-ins which you won't need for just recording your bass, and AFAICT it requires a software app to control some features, which some people don't like. So, buying brand-new, I'd go 2i2.
REAPER is indeed an excellent tracking/mixing DAW. I've been using it since late in version 4, they're into version 6 now. Your $60 personal-use license (assuming you aren't making money from using the software, at least not yet) gets you free upgrades through two major versions, which historically speaking is about 7 years' worth of updates. It also gets rid of a nag dialog that is cleverly designed to require that dialog window to have focus for 5 seconds before you can clear it. That's basically the only limitation on the unlicensed version of the software; you can use any feature, no time limit, no artificially lowered limits on tracks or plugins.
Apart from these two tools, all you need is to connect the amp to the interface, which can be accomplished with a $15 20' mic cable (XLR to XLR). That will use the "transformer balanced line out" of your PortaFlex. Bear in mind this output is taken between the preamp and power amp, so you won't get the sound color of your tubes or of the speaker cabinet.
To capture these elements of your full rig's tone, you'll want to put a microphone in front of the bass cabinet. Any old mic will work in theory, including ones popular for guitar cabs like the workhorse SM57, however you may lose low-end. Mics that are specifically recommended for bass guitar tend to be those with a combination of good low-end frequency response (<40Hz) and high SPL handling (>130dBA; keep in mind the mic will be mere inches from the cone, so the SPL it's subjected to will be 30-36dB higher than your own ears' volume level from 6 feet away).
The AKG D112 is an industry standard for bass instruments including kick drum and bass guitar cabinets, but it's a $200 microphone at retail so kind of a pricey piece of kit for someone just starting out. It's also deaf as a post (a natural tradeoff of its high SPL handling), with a sensitivity of -55 dbV at 94dBA. That's a little better than some other well-liked dynamics (the popular SM7B vocal mic is -59dB), but you might have trouble boosting that to mixable levels using only the Scarlett's 46dB of preamp gain. You may want to find a preamp of some kind; there are tons of options, but I wouldn't start looking along these lines until you're sure you need one, and that you want to keep building your home studio. You may be better-served at that point by buying a small mixer with good preamps instead of a one- or two-channel mic preamp unit.
