First of all you have to understand what is meant by "millionaire". There are dollar millionaires and sterling millionaires for instance. There are paper millionaires and cash millionaires.
In any case...
There are a few simple rules. First of all, always, without exception, pay yourself 10 per cent of what you earn.
People earn a sum, there are statutory deductions, and they are left with "take home pay" or whatever's left -- they then pay the bills, the mortgage, insurance, more taxes etc -- they may even treat themselves by buying something... but they always forget to pay themselves! The trick is to always take off your 10 per cent straightaway. Any debts should be paid from the next 10 percent, and the rest is for everything else as usual. Never ever spend the money you pay yourself, just let it accumulate, invest it, earn interest with it -- and borrow against it.
The second rule is never use your own money when you can favourably risk someone else's. If you need money for a venture, borrow against the equity on a property (difference between what you paid for it and it's present worth).
The third rule is always pay back your debts. If you have never had debts you will not be better off. It is better to have had debts and paid them off with interest -- you become an attractive prospect.
The fourth rule is to buy cheap and sell dear. Always. And borrow to do this. You can also control supply by stockpiling or otherwise manipulate demand.
The fifth rule is to cross markets. You MUST cross from one marketplace to another; it is the only way to make money fast. For example, buy a house, wait and sell it when prices have risen -- use the money to buy another house in a completely different market (different area) because all the ones in the same market will have risen by the same amount. Another example is to buy something in a car boot sale and sell it on eBay (different market) or auction -- or in a shop window. If you can buy something cheaper abroad and sell it at a high price here, then do so.
The sixth rule is to copy the market. Never be put off by others. If a restaurant opens, then open a restaurant next door and match their prices. If there is sufficient demand for one, there will be for two. You can cash in on their advertising -- people will go to you if they are busy.
So: by saving and investing you can become eligible to borrow large amounts to buy something than can rise in value (such as property) -- sell and cross markets, pay off debts and re-invest and borrow more, buy cheap and sell high, interest on loans must be less than interest earned.
Within a very short time you should be able to declare yourself a paper millionaire and a little while later on, a cash millionaire.
The BBC ran a series of TV shows where two guys simply bought houses, did them up, and sold them for profit -- re-investing and moving to different areas around the UK. Within one year they had achieved a million pound house for charity.
The luck element merely affects the length of time before you are a millionaire -- if you are unlucky, it will only take longer. A luck break is when you get something really cheap or a better than expected sale or rate of interest.
Good luck from someone who knows ;)