Basically, never claim exempt on your W-4. The withholding system is set up so that, if you earn under the amount to owe tax, no federal tax will be taken out of your pay, and if you start to earn over the limit, the appropriate tax will be withheld. In other words, it's self-adjusting. Claim the correct number of allowances per the worksheet.
At $15,000 of gross income, your taxable income will be $3000 and your income tax owed is $300, assuming this is all from W-2 employment and not self-employment, and assuming nothing else changes in your tax situation. You could claim exempt once you reach this amount -- the IRS will not penalize you as long as the amount of money you pay into the system is equal or more than what you end up owing. But be careful; if your situation changes (you start working more hours, you get married, you start a side-job or take a second part-time W-2 job) and you don't adjust your W-4, you could end up with a surprise tax bill.
Withholding limits, exemption amounts, and tax percentages may be completely different for your state income tax. Claiming exempt on your W-4 may also make you exempt from state withholding (unless you file a separate state withholding form claiming something different) and that might be the wrong answer even at that low of income.
