Watching my grandchildren in my home. How do I claim the income and what can I use as deductions for tax purposes?

Report the income as self employment and deduct any associated expenses such as food and toys you provide. You will pay income tax at your marginal tax rate as well as 15.3% self employment tax. 

To report your self employment income you will fill out schedule C in your personal 1040 tax return and pay SE self employment Tax.  Here's a Schedule C  https://www.irs.gov/pub/irs-pdf/f1040sc.pdf

You can enter Self Employment Income into Online Deluxe or Premier but if you have any expenses you will have to upgrade to the Self Employed version or use any of the Desktop CD/Download programs.

How to enter self employment income

https://ttlc.intuit.com/community/self-employed/help/how-do-i-report-income-from-self-employment/00/26653

You will need to keep good records.  You may get a 1099NEC at the end of the year if someone pays you more than $600 but you need to report all your income no matter how small and if you don't get the 1099NEC.  For the future, you should use a program like Quicken or QuickBooks to track your income and expenses.  There is a QuickBooks Self Employment bundle you can check out which includes one Turbo Tax Online Self Employed  return....

http://quickbooks.intuit.com/self-employed

You use your own records.  You are considered self employed and have to fill out a schedule C for business income.   You use your own name, address and ssn or business name and EIN if you have one.   You should say you use the Cash Accounting Method and all income is At Risk.   

After it asks if you received any 1099Misc or 1099NEC it will ask if you had any income not reported on a 1099Misc. You should be keeping your own records.  Just go through the interview and answer the questions.   Then you will enter your expenses.

Self Employment tax (Scheduled SE) is automatically generated if a person has $400 or more of net profit from self-employment.  You pay 15.3% SE tax on 92.35% of your Net Profit greater than $400.  The 15.3% self employed SE Tax is to pay both the employer part and employee part of Social Security and Medicare.  So you get social security credit for it when you retire.  

The SE tax is already included in your tax due or reduced your refund.  It is on the 1040 Schedule 2 line 4 which goes to 1040 line 23.  The SE tax is in addition to your regular income tax on the net profit.

Here is some IRS reading material……

IRS information on Self Employment

http://www.irs.gov/Businesses/Small-Businesses-&-Self-Employed/Self-Employed-Individuals-Tax-Center

Pulication 334, Tax Guide for Small Business

http://www.irs.gov/pub/irs-pdf/p334.pdf

Publication 535 Business Expenses

http://www.irs.gov/pub/irs-pdf/p535.pdf