When recording direct, you will need a line-level input for your amplifier. Most standard soundcards have one, but the quality of the pre-amps and A/D converters is usually nothing to write home about, so you'll probably want a separate piece of gear if you want good results.
You shouldn't need a DI box, unless you plan to keep the amplifier a long distance away from the recording input and/or use a balanced XLR plug for the inpunt. Nor will you typically need a mixer, unless you intend to record several signals at once. What you may want to get, however, is a USB or FireWire recording interface.
The interface doesn't have to be particularly complicated or expensive. If you're simply going to be recording your guitar, a simple two-channel one will do. Check the amplifier's manual and see what the output level is - the two standards are -10dBu and +4dBu. You can find this information in the Specifications section for the headphone/record out. If there is no information, it's probably a good bet that the output is -10dBu. Make sure that your interface supports the appropriate output level. Most units available these days should handle both, no problem.
It may be the case that the amplifier offers a choice of output levels, in which case you should choose +4dBu (don't forget to set your interface for that as well). Plus, check your manual for information about making connections and the like. You'll probably need a Y-cable that will plug into the 1/8" TRS headphone/recording output and split the stereo signal into two 1/4" TS jack inputs (that you'll plug into the two line inputs on your interface, for the Left and Right sides of a stereo recording).
Check if you have to switch between different output settings for headphones and recording - headphone outputs generally have filters on them to optimise the output for listening, not to mention the fact that the signal is amplified in order to drive the headphone speakers. Both of these features are unnecessary when recording direct and can have a detrimental effect on the recorded sound. If there is no switching option, you may have to tweak the settings on your interface (reduce input gain, adjust the EQ etc.)
Lastly, you will need to set up your recording software for monitoring your recording. If you're recording through an external interface, you'll typically have to connect your speakers to the interface outputs for this purpose, or plug headphones into the interface if it has a headphone out. Furthermore, when monitoring a recorded signal, you'll typically find that there's a delay between you playing a note and the sound coming from the speakers/headphones (this is called latency). The latency will depend on the settings of your software and - indirectly - on the parameters of your computer. Reducing latency will put a greater load on your system and you'll find you can only bring it down so low, before you start getting dropouts and general playback trouble. If you have a 'direct monitoring' option on your interface - which allows you to monitor the signal entering the interface's input, as opposed to that which has gone through your recording software - it might be a good idea to use that. (Some software - ProTools, for example - also has a 'low-latency monitoring' option, which bypasses most signal processing in order to offer better performance).