I and my employer made less than $3250 contribution to my HSA in 2013. I heard that I could still make contribution for 2013 before 4/15/2014. If this is true, and I deposit my personal money to HSA now to max up contribution for 2013, will that new contribution still be tax-deductible?
Yes, you can make contributions to your HSA for 2013 until April 15, 2014. However, when you make the contribution, you need to explicitly tell the financial institution that your deposit is for tax year 2013, as they will be reporting to the IRS your total 2013 contributions after April 15.
If you haven't filed your tax return for 2013 yet, you can wait until you make your final HSA contribution for 2013, and then file (before April 15). Otherwise, if you have already filed for 2013, and you make another HSA contribution for 2013, you'll need to file an amended return in order to claim the additional deduction.
One more piece of advice: Don't wait until the last day to make this contribution. Make your final contribution at least a week early. Each year here on the site we get some questions from people who made an HSA or IRA contribution on April 15, only to be told by the bank that they missed some internal bank deadline by a few hours. There is no need to wait until the last day; don't cut it too close.