First, a disclaimer: This is certainly not the only way to succeed; probably it isn't even the best way. But I know I was totally lost the first few times I tried this and now I have it down to a science, so take what tips you like and use them as you will.
SEVERAL DAYS BEFORE
1. Make sure your house sits relatively close to the street and is easy for Sims to navigate. Remember that stairs are like Sim kryptonite. Relatedly, try to have at least 4 or 5 good-sized distinct rooms on the ground level of your Sims' house. (If you have an eat-in kitchen, you might consider putting up a wall and archway, etc.)
THE DAY BEFORE
2. Decorate! Flooring, walls, and lights are the essentials. (I always forget to give my Sims lights until it's time for the headmaster to come over. Good thing they can read in the dark...) Then plaster the walls with paintings. I like paintings best because they don't take up any floorspace, and there are quite a few high-scoring 1-tile paintings, so you should be able to fit them in between windows and doors. To test your decorating abilities, select a teen/adult/elder Sim to wander around and make sure their environment motive stays in the high green.
It is not necessary to decorate anything other than the ground floor of your Sims' house, for reasons that will become clear in a moment. You also do not need to decorate really tiny rooms.
2b - The Quick and "Dirty" Decorating Method: Buy one high-scoring statue (in the description, it will say "Environment 5", "Environment 10", etc). Buy Mode is still accessible during the headmaster visit, so you can pick up the statue and move it from room to room as your Sim gives the tour, thus giving you a higher tour score without tedious decorating or spending tons of cash. In this case, statues are better than paintings because you don't have to spin around looking for a blank wall. Just plop that sucker down in the middle of the room.
3. Have the teen/adult/elder Sim with the highest cooking skill start cooking as soon as "Serve Dinner" (rather than "Serve Lunch") becomes available. Try to have them cook something nice, like pork chops or salmon or turkey or lobster thermidor. (I've gotten 45 points from lobster thermidor alone, seriously.) If they burn it, make them try again until they get it right. Once they succeed, put the leftovers in the fridge and don't let anybody eat them.
(Step 3 only works if you have Seasons, I think. If you don't have the ability to "Put away leftovers", don't worry. I'll address that in a minute.)
THE DAY OF
4. "Invite headmaster" is its own tab on the phone. I am reasonably certain that teens can call the headmaster; I don't know if children can. The headmaster will arrive at the first 5 p.m. after your call. (So, if you call at 4 p.m., he will arrive an hour later. If you call at 6 p.m., he will arrive the next day.)
Don't bother inviting the headmaster if no child/teen has a grade of C or higher. (I am not sure about C minus; I think that might be too low.) This does mean that you can invite the headmaster as soon as your child ages up, since they age up with a C.
(For my readers playing a non-English game: The school performance bar has to be at least halfway full.)
5. As soon as your children get home from school, PUT THEM TO BED. If they're not sleepy, confine them to an out-of-the-way corner with a skill book or something. The goal is to have them doing something that they will keep doing for 4-6 hours. Lock them in the basement if you want; I don't care. The headmaster doesn't care either.
Do the same with other extraneous Sims. You should have no more than 2 teen/adult/elder Sims up and about by about 4:55 p.m. (Sim time, of course.) One of these Sims will henceforth be referred to as Tour Guide, and the other as Chef, for reasons that will become obvious in a moment.
6. The headmaster arrives at 5:00, as I mentioned above. Have Tour Guide go greet him. Click on the headmaster and select "Entertain > Give Tour." Have Tour Guide "Go here" to whatever room in the house is prettiest. Click on the headmaster again and select "Show Room." Then repeat until the headmaster thanks you for the tour. (This usually happens once you get about 45 points, which usually takes me 4 or 5 rooms, though I think I've gotten it with as few as 3.)
This is the part where my advice to make the ground floor as nice as possible comes in. DO NOT TAKE THE HEADMASTER UPSTAIRS. I once had a headmaster stuck upstairs for hours. I don't care how pretty the rooms are up there, it's not worth it.
I also generally avoid bathrooms or other tiny rooms because of potential route failure.
If you have shown all the rooms you want to show, you can also click on the headmaster and select "End Tour."
7. While all this touring is going on, have Chef pull out the leftovers from back in Step 3. If you don't have Seasons, have Chef cook. If Chef burns food on the first try, have Tour Guide kill time by serving the headmaster espresso or bar drinks. (I'll cover these topics later on.)
When the food is set out, click on the headmaster and select "Call to Meal." The headmaster will sit and eat for what seems like forever, and the food score is all or nothing...if you time out when he has 1 bite left, TOO BAD FOR YOU. So try to serve food fairly early on in the evening. If time is running short, don't let your other Sims sit and talk to him while he eats; that slows him down and the schmooze points are negligible.
8. If you want schmooze points, do NOT use the "Schmooze" interaction. Just build relationship as you would with any other Sim...talk, joke, etc. I don't usually get a lot of schmooze points, so I don't have any other tips for this category.
9. You can get bonus points for three things: Espresso, drinks from the bar, and hot tubbing. To get the espresso points: Buy an espresso machine. (Obviously.) Have your Sim (usually Tour Guide) "Serve Espresso" rather than "Make Espresso." (The former produces 2 cups; the latter only 1.) Have your Sim drink one of the cups. The headmaster usually takes the hint and drinks the other cup.
Drinks at the bar are similar. Serve, partake, wait for the headmaster to catch on and join you.
For the hot tubbing, get in and ask the headmaster to join you. (He might also join you on his own; I don't know.) I dislike this bonus point opportunity, though, because it can lead to lots of route failures if you're not careful and that's just a pain.
10. Once you top 90 points, the headmaster will say "X, Y, and Z have been accepted into private school", shake hands with one of the parents, and leave. (People can get more than 90 points, of course. Say you have 85 points and then the headmaster eats lobster--you'll end up with about 130 points total because of getting 45 in one shot.)
I find that a good tour and a good meal can be enough all by themselves, but if your Sims can't cook lobster or something it's easy enough to nudge the points over the top with espresso or drinks. As I mentioned above, I almost never use schmoozing or the hot tub.
Using this method, I can get private school admittance in less than 2 Sim-hours, and I think the headmaster scenario lasts for 4 or 5 hours. It's been a while since I've timed out, so I'm actually not sure.
I hope this post is helpful to somebody (is anybody still new to Sims 2 in 2013?) but if not I had fun writing it anyway.