I was an au pair under J-1 visa last year, I got married to a US citizen in September last year. I do not have the green card now.How can I do tax forms?

It depends -

As a J-1 visa, you will be exempt from the Substantial Presence Test for the first 2 (if non-student) or 5 (if student) calendar years that you are in the US. If this applies to you then you are considered a non-resident aliens for US tax purposes.

Click this link for more information from the IRS on US Tax Guide for Aliens

You will need to file a Form 1040NR to report your US source income. TurboTax does not support Form 1040NR. Here is a link to the IRS website for Form 1040NR, U.S. Nonresident Alien Income Tax Return

However, since you are married to a US citizen, you can file a joint return with your US spouse but you will need to recognized your worldwide income on your jointly filed Form 1040. (Your occupation can be listed as "au pair" or "Nanny".)

Click here for more information about Claiming Non-Citizen Spouse and Children On Your Taxes.

In order to file jointly, if you do not have a Social Security Number (SSN) or Individual Taxpayer Identification Number (ITIN), you will need to apply for one. To apply for an ITIN, you would use Form W-7 and would need to attached your tax return to this form (see below).

Here are some additional steps for filing married filing jointly if you needs an ITIN. 

• leave your SSN blank and print a paper copy of your tax return.
• you will may need to use an SSN holder number to get through the health care section (try 999-11-9999).
• skip error check section if only error is missing SSN for you.
• under the file tab, you will need to select "file by mail" and print a copy of your federal income tax return.
• attach a statement, signed by both spouses, to your joint return for the first tax year for which the choice applies (Nonresident Spouse Treated as a Resident). It should contain the following information:
•  A declaration that one spouse was a non-resident alien and the other spouse a U.S. citizen or resident alien on the last day of your tax year, and that you choose to be treated as U.S. residents for the entire tax year.
• The name, address, and identification number of each spouse.
• apply for your Individual Taxpayer Identification Number (ITIN) using Form W-7 (Click ITIN).
• attach a copy of your Form W-7 (along with the necessary supporting documents) to your tax return (For the supporting documents, one option is for your NRA spouse to get a certified copy of his/her passport which will need to be included with your tax return filing.) Click here for IRS website  ITIN - Frequently Asked Questions .
• mail your tax return along with your declaration statement, Form W-7 and supporting documents to the IRS. According to the IRS, because you are filing your tax return as an attachment to your (ITIN) application, you should not mail your return to the address listed in the Form 1040, 1040A or 1040EZ instructions. Instead, send your return, Form W-7 and proof of identity and foreign status documents to:
  
       Internal Revenue Service
       Austin Service Center
       ITIN Operation
       P.O. Box 149342
       Austin, TX 78714-9342 
  
  You may also apply for an ITIN using the services of an IRS-authorized Acceptance Agent or visit some key IRS Taxpayer Assistance Center in lieu of mailing your information to the IRS in Austin.

• State filing - You will want to file an extension for your state until you receive your ITIN. This is because the application for your ITIN is a federal filing. Most state will not process your return without either a SSN or an ITIN. (However some states will accept state filings with W-7 attachments. Please contact your state for more information using the State Department of Revenue Contacts and ask if your state will accept a state tax return with a Form W-7 attached).  
  When the IRS assigns you an ITIN, update your returns (both federal and state) in TurboTax with this new information and then you should be able to e-file your state return.
  Because many states give an automatic extension when you file the federal extension Form 4868, just file a federal extension even if you don't think you will need it for your federal tax return.  But check with your state's Department of Revenue to make sure that your state doesn't have a separate state extension filing.