The bare-bones way is to use the microphone or the line-in. I found the line-in to be a better choice, but either way, you need to reduce the amplifier volume to avoid clipping. This volume level will be pretty low, and it is specific to your equipment. After you adjust the amplifier volume, you can then adjust the overall volume on the computer. Audacity is basic recording software which can help you visualize the input levels and is open source.
The less bare-bones, but still free way is to plug the bass directly into the input (without amp nor effects), use ASIO4ALL software, a VST Host application such as VSTHOST, and some amp simulators (google VST AMP SIM). You can set a program like Audacity to record "what you hear" on the sound card, and in this manner you can record the audio after it has passed through the VSTHOST application. (TLDNR; ASIO4ALL is a driver for your sound card; VSTHOST is a container program that allows you to load VST effects and chain them together; Audacity is an audio recording and editing application)
A good recording application which is not freeware, but has no usage limitation before licensing is REAPER. Reaper is a full-on digital audio workstation and is a VST host application. If you use this, you don't need VSTHOST, just the amp sim and effect VST programs. (TLDNR; ASIO4ALL; VST Effects; REAPER to record and host VST effects)
The advantage of recording the raw signal and applying the amp and effect sim(s) to it at mix time is that you can add or modify the total effects chain at will long after the performance is recorded.
AFAIK, all of the above is free (except Reaper which is free to try) and freely available. Once you get a sense of the process and confidence that your equipment can handle the task, you can make a decision about where to spend the money to improve upon it).
Note that ASIO4ALL is there to reduce driver latency, without it, the delay between what you play and what you hear can be upwards of 20ms (or more) and this is pretty much impossible to work with. Some sound cards/systems already have an ASIO driver bundled (in which case ASIO4ALL is not needed), but most don't. I personally found in my early experiments that increasing the sample rate of my sound "card" (onboard Realtek audio) to 96khz instead of the standard 44/48 helped reduce latency when using the ASIO4ALL driver.