Philostang and JSbumed have already brought up good points... A few things to think about if you're making a serious bid at getting into TT with the intention of actually doing well, though.
1) IT'S NOT A RACE!!! Yes, it is "legal" to go through the corners door-to-door in TT, however everybody knows that if you do, BOTH cars are going to be slower as a result, and thus have "blown laps." If you make a habit of driving that way in TT, you WILL wind up getting a bit of a talking to from a variety of people. There are many that take the competition end VERY seriously, and if you screw with them (on track), it will wind up becoming an issue.
2) If you've got some track experience, think about what the real drive is behind TT. Frequently, people go out for three laps in a session. Warm-up, a hot lap, and then a cool-down and off. If you think it's going to give you a ton of "no rules" track-time, think again.
3) If you've only got a little bit of track experience, realize that the guys in TT are F-A-S-T! Frequently, the TT crew sets faster lap times than the equivalent race group does. Different mentality, shared skill set, different approach to "the game." If you are prone to making more than one or two tiny errors on a given lap, and can't regularly nail at least two or three solid laps per session, then you're not ready to dive into the deep end. At a minimum, you should completely understand the concepts of momentum-style driving (even in a Viper) and tire slip-angle, and how to use them to your advantage. Know the difference between street rubber, and R-compound or slicks, and know how to drive them to get the most grip out of them. You will also need to be completely comfortable in an open-passing environment, and not even have to think about trail-braking, heel-toe downshifts, offline cornering (at traction limit), traffic pacing, etc. If those aren't second-nature, then you're not ready.
4) The level of competition can sometimes be FIERCE!!!! In 2010, I lost a season championship by 0.001 seconds... It doesn't get much closer than that.
All of the above said, as a general rule, the TT drivers are absolutely outstanding in terms of mental focus and car-control skills. There is also an assumption that if you're out in that group, you can hang; if you can't then it potentially becomes a safety issue. I don't want to make it sound "elitist" but honestly, not everybody is cut out for TT. From a NASA perspective, there's no difference in the "rules of engagement" between HPDE-4 and Time Trial, other than TT is considered a competition group, and HPDE-4 is not. If all you're looking for is the open-passing environment, and couldn't even consider coming in early from your session (I want the track time!!), then my advice is to consider staying in HPDE-4. If, on the other hand you crave the competitive environment, but don't want to cage your car and go racing, then TT may well be the perfect environment for you.
Philostang and I are both NASA instructors, and JSBumed is both an instructor and a Time Trial Director, so we all speak from a fairly similar viewpoint. The actual mechanics of getting into TT are pretty simple, really. Present yourself to the regional TT director, with a filled-out Time Trial Provisional License application, and then follow their direction on how to proceed from there. If they don't know you, there's a good chance that they'll toss an instructor in with you for a session as a check-ride. Different regions may have different specific things they'll look for in the check-ride, or different drills and tests they'll run, but be prepared to do a lot of things in a very short period of time that will stress you in a very high-speed environment. We'll be looking for general level of comfort, basic driving skill level, familiarity with and appropriate use of advanced skills, ABSOLUTE situational awareness, and appropriate pace. We could care less how fast (or slow) you are, as long as you can wheel the car safely. Over-drive it, spin it, go off, etc., and the check-ride is done, and not with a happy result. Assuming all goes well, at the end of the weekend, the director will counter-sign your provisional application, then you fax/email it to National, and shortly thereafter you'll get your hard-card license in the mail. Done. That license is portable to all NASA regions throughout the US.
Now, as for how long it takes the "average" student to get to the skill level necessary to safely compete in TT? The hard, numerical answer is: "It depends." One of the biggest things about wanting to hang out in the HPDE ranks is experiential learning. Is it possible to be a "natural," blow through the ranks in a few weekends and then get a TT license? Absolutely. It will actually be no problem, either, until you're out in the TT group when the heavens open, and you suddenly discover that you have NO idea what a "rain line" is used for, how to find it, or how to drive it. OR you're out there blowing through a corner at 1.4G and 95MPH, and when you pick your head up at corner-exit, you discover a car STOPPED right in front of you. (Hint: your head should have been up before turn-in, not at exit!) You might be the fastest thing on the planet after Cole Trickle, but if you have no idea what to do when they're waving a red flag at you, things could get ugly in a hurry.
I guess what I'm trying to say (in my normal rambling, long-winded manner) is that taking the jump into a competition group (TT, W2W, Hill Climb, whatever) isn't one to be taken lightly. Make sure you're ready for it, read up on it, study it, get to know the players, learn the procedurals and concepts, and you'll not only go further faster, but you'll have more fun doing it.
Oh, and for spending points in TT: Do as much of the "free mods" list as you reasonably can. Get your car right down to minimum weight (Hint: The first five lbs under min weight are free, at least through 2013). Spend as many points as you can on sticky rubber. Spend as much of the remainder as you can on WIDER sticky rubber. Use whatever you have left over on a CAI and tune, then dampers and springs, and if you have any points left over after that, either get stickier or wider rubber, THEN go for weight reduction. In the TTB-TTF classes, aero is a fools game, the points just cost too much for the benefits. In the TT# classes (TT1, TT2, TT3) aero isn't just a good idea, it's essentially mandatory if you want to run up front.
Like everything with TT, MAXIMIZE your points spent. If you're going to do a wing, don't do a GT500 lip spoiler, go to APR, Good Aero, G-Stream, etc and get a REAL wing that actually does something. Yes, the G-Stream wing is $1300, PLUS the cost of the mounts, but in TT terms, it "costs" the same (points) as simply deleting the factory wing. If you do dampers, look at Sachs, AST, Penske, not at Koni yellows or D-Specs. Again, 3x the price, same points cost, and arguably 2x the "value."
Also, as was brought up before, if you want to be competitive, you have GOT to maximize your setup right to the limit of the rules. If you build around TTB, though, I bet you'll suck in SCCA STU. And vice-versa. It's kind of like building a combo street/race car: it will do neither well.
ddavidv: Welcome to the ranks of the clinically insane! Who else would climb into the right seat of a car driven by a complete stranger, and yell at them to "go faster!" ;-) I've been instructing for a few years now, and I STILL have to say the most memorable moments in any given weekend are from the right seat. That "Oh!" you hear over the intercom when they "nail" a perfect corner is all the payoff we need. Well, that and that delicious free NASA water!
