Start with the disappointing bit: unless you live somewhere high, you cannot train for altitude. No amount of running teaches your body to cope with thin air. What training does is make the walking cheap, so you arrive at the top camp with energy to spend on the only hard part.
Train the thing you'll actually do
Which is walking, uphill, for hours, carrying a small pack. The best training is long hill walks, and there isn't a close second. Build up to a six- or seven-hour day with 700 to 1,000 metres of ascent, and do it on consecutive days at least once, because the mountain doesn't give you a rest day after a hard one.
A rough shape
Three to four months is enough for most people starting from reasonably active. Two walks in the week, one long one at the weekend, and some stair-climbing or step-ups if you have no hills nearby. Add a couple of strength sessions for legs, hips and core, mostly to protect your knees on the way down, which is where people actually get hurt.
Walk downhill on purpose
Descent is the part nobody trains and everybody complains about. You lose around 4,000 metres of height in a day and a half at the end, much of it on loose scree, and quads that have only ever gone uphill will let you know about it. Put deliberate descent into your long walks.
Break in your boots, and your pack
Every long walk should be in the boots and socks you'll wear on the mountain, with the daypack you'll actually carry, loaded with water, layers and snacks so it weighs what it'll weigh. A blister on day two is a long week. Kit is covered properly in the packing list.
What not to bother with
Altitude tents and hypoxic chambers have thin evidence behind them for a trip like this, and they cost real money. Heavy gym training does little for a long slow walk. Running is good general fitness but it trains the wrong pace and the wrong muscles, and runners are among the most likely to walk too fast on day one.
Being very fit isn't the advantage you'd think
Fit people feel strong low down, so they push, and pushing is what gets you in trouble. Altitude doesn't care about your parkrun time. The people who do well are usually the ones who are content to walk slower than they need to for six days.
The last two weeks
Taper. Keep moving, drop the long days, and arrive rested rather than peaking. Turning up tired from one last enormous training weekend is a more common mistake than turning up undertrained.
If you live somewhere flat
Stairs. Genuinely. Find a tall building or a stadium and go up and down it with a loaded pack, which trains the climb and, more importantly, the descent. A treadmill at maximum incline works for the uphill half. Long flat walks still build the endurance you need, so don't skip them just because there's no gradient; four hours on the flat is worth more than one hard hour in a gym.