Income Tax is calculated by the same method for weekly pay as it is for monthly pay, except that your annual personal allowance (£7475) is divided by 52 (instead of by 12) to spread it across a weekly basis. That's roughly £144 per week that you can earn each week before you pay tax.
So, for example, at the end of the 10th week in the tax year you'll have accrued £1440 tax-free allowance. The employer looks at the gross amount which you've earnt up to that point, deduct the £1440, and calculate 20% of what's left. That's the total amount of tax you should have paid to date. Then he looks back at how much tax you'd already paid by the end of week 9. The difference between those two figures is how much tax you now have to pay.
National Insurance is calculated weekly (rather than on a cumulative basis. You don't pay anything on the first £110 you earn but you pay 11% of everything after that. (If you're lucky enough to earn over £844 per week you pay at a much lower rate, of 1%, on whatever you earn above that figure).
The term "cash in hand" is often used when referring to 'under the counter' payments, without any deductions for tax or National Insurance, to workers who don't appear on the company's books. If that what your employer is proposing, it's to be avoided at all costs. (It's illegal and, because you're not officially an employee you'll have no rights to paid holidays, or any other employment rights). However I suspect that your employer is simply offering to pay you in cash (after making the necessary deductions) rather than paying the money directly into your bank. If so (unless you really need to have 'folding money' put directly into your hand), BACS is probably the preferred option, since the regular flow of money into your bank account will help with any credit applications you might wish to make.
Chris