If my previous employer made me pay back moving expenses because I didn't work there long enough, is the amount I had to pay back tax deductible?

It's complicated.

Yes, if you paid back income that was previously taxed.

No, if the moving expenses were tax free.

Most company paid moving expenses are a mixture of the two. The company is allowed to pay certain moving expenses on a tax free basis; e,g. your travel and the cost of moving your household belongings(things that are normally tax deductible). Anything else they pay is suppose to be imputed as income on your W-2. 

How to claim repayment of taxable income

This situation is  described in IRS Publication 525. pgs 34-35.  You can take a  misc itemized deduction, on line 28 of schedule A (not subject to the 2% of AGI threshold) Or you can take a credit. The credit is computed by refiguring the tax return from the previous year as if the income had not been received. Then the difference in tax is claimed as a credit on the current year's return.  In the forms mode (the forms mode is not available in the on-line versions of TurboTax[TT]), you can use the line 73 smart work sheet to enter the credit amount on line 73; select item D, claim of right under IRC 1341 for repayments. TT will  enter "I.R.C.1341" on the line next to box d on line 73 of form 1040. TT does not do the calculation or compare it to the alternate deduction.

So, the taxpayer  has the option of either claiming the credit or deducting the repayment as a miscellaneous itemized deduction, whichever provides the greater benefit. But, you cannot file an amended prior year return. If the amount is $3,000 or less, only the  miscellaneous itemized deduction is allowed and is subject to the 2% of AGI threshold; it goes on line 23 (instead of line 28) of schedule A.

https://www.irs.gov/publications/p17/ch12.html#en_US_2016_publink1000172015