I made a contribution to my Roth IRA and did not know there is an income limit that precludes one from making contributions to Roth IRA if one is over that limit. Now that I discovered this, I have recharacterized the amount as a non-deductible traditional IRA. Am I still eligible to use the backdoor and recharacterize the traditional IRA as Roth IRA effectively undoing everything to begin with? How should I file my taxes if I do this?
You can certainly (belatedly) use the backdoor (assuming you have no money in pre-tax IRAs of course).
Your terminology is wrong. A "backdoor" Roth IRA contribution involves making a Traditional IRA contribution and then making a conversion from Traditional IRA to Roth IRA. You recharacterized a Roth IRA contribution into a Traditional IRA contribution, so now you have the first step. Now all you need to do is convert the Traditional IRA into Roth IRA. You can certainly do this now (or any later time, but earlier the better).
Note that because of the time gap between the contribution and the conversion, there may be earnings made during this time, which will need to be taxed on conversion, which would not have happened had you done the two steps (contribution and conversion) together initially.
Not this year. Here's what the IRS FAQ on recharacterizations (http://www.irs.gov/Retirement-Plans/Retirement-Plans-FAQs-regarding-IRAs-Recharacterization-of-Roth-Rollovers-and-Conversions) says:
Is there a minimum waiting period to reconvert the money to a Roth IRA following a recharacterization?
Yes, if you recharacterize all or part of a rollover or conversion to a Roth IRA, you cannot reconvert the amount recharacterized to the same or another Roth IRA until the later of:
30 days after the recharacterization, or the year following the year of the rollover or conversion.
The waiting period to convert applies only to amounts you recharacterized. For example, you can convert amounts from a different traditional IRA to a Roth IRA immediately.
Although the FAQ and the above quote from it only talk about rollovers and conversions, not contributions as is the case in this question, if you look at Publication 590-A, in particular the Recharacterizations section, it talks about recharacterizations of contributions as well.