This can take a while, but is pretty straight forward. If you are following the road map, do this on your second map after building the necessary tube stations for Subway Mall Expansion. The easiest way to do this is to start with $4-5 billion and open the Market menu. Use to go to the Stock Market and use the D-Pad if necessary to select Buy Shares. Change shares purchased to the maximum of 10,000,000.
To make money on stock market, as in real life, you want to buy low and sell high. The advantage of this simulator is that it has a cyclical nature that is only psuedo random. You lose 10% of the total price for each transaction, so you need a fairly wide margin to make money. I recommend anything under $8 as the purchase indicator, and anything over $12 to sell. While the game sometimes drops as low as $5 or $6/share, and goes as high as $20, that seems to be pretty rare. The general cycle was $7-14, so those cut offs work well to frequently be able to make money.
Anytime a stock price is in the purchase range, buy 10,000,000 shares. Anytime it is in the sell range, sell all the shares you have in that stock. It is easier to pause and scroll through all 50 or so stocks available so that you have the time to purchase or sell all appropriate stocks. Generally what will happen is that you will end up owning 10,000,000 shares in each company, wait a while, then sell each off as it rises above $12. You then wait until the market drops again, rinse and repeat. Not a fast process by any means, but it does work and depending upon the market cycles, should take less than 2 hours.