How can I claim my taxes by getting paid under the table?

There isn't a legal way to do that unless you use turbo tax or hire some sort of accounting firm of that sort (turbo tax.. h&r block... ) who will fill it out for you.

You should wait till you get all the Forms like W2s and 1099s. If you file based on last pay slip, you may have to file amended tax return if there is any different figure in W2 or 1099.

You CAN claim wages that are not documented with a W-2, but if you do so, expect to write a rather large check when you file the return, because there will be NO withholding to offset your tax liability.