Yes, you can still make a 2013 Roth IRA contribution before the tax filing deadline. But be sure to check the income limitations for making Roth IRA
contributions. High earners (defined differently for different filing status)
are not permitted to make Roth IRA contributions. If you are expecting
to get a refund on your 2013 tax return, you can even file early and use
the refund toward your 2013 Roth IRA contribution. Note that Roth IRA
contributions are not declared on your income tax return and so
it is not necessary to file an amended return after making the Roth
contribution (see this question, pointed
out by @JoeTaxpayer, for details on how the IRS knows that you have
made a Roth IRA contribution). Indeed, if the refund does not come in
time and you choose to not make a Roth IRA contribution for 2013 after
all (or decide to make a reduced contribution), no
paperwork needs to be done by you.
You can also make Traditional IRA contributions after filing your
tax return but before the deadline, and use the tax refund from the
2013 tax return towards the 2013 IRA contributions. However, if the
tax refund is not received in time, and you choose to reduce your
2013 Traditional IRA contribution (or not make it at all), then
an amended return is required (and you owe tax or will get a
reduced refund) because
you have already declared that you will be making a Traditional
IRA contribution (and deducted that amount) when you filed your
income tax return early and computed that you overpaid and will
be getting a refund.