Who grants pardons for felony convictions?
If you were convicted of a felony at the federal level you actually need a pardon from the President of the United States. If you were convicted at the state or county level you need a pardon from the Governor of your state.
The real challenge, though, is getting your pardon request to one of those levels. Governors and Presidents are really busy (surprise, surprise), so they have boards of people that review requests all day. Most requests are denied - you need to have a really good reason for requesting the pardon (also known as clemency in most places).
Where do you start?
First of all, you need to understand that a lot of inmates sitting in jail are sending pardon requests to the Governor (or the President, depending on their jurisdiction). If you just try to sit down and write a sob story explaining why you need clemency, you probably aren't going to get past the first gatekeeper.
So how do you make sure your pardon request is taken seriously? You will probably need a good attorney. Sure, they can help you draft your letter in a way that won't come off as whiny or desperate (which so many pardon requests are) and make it sound sincere, but more importantly, they will help you get it in front of the right people.
How to find the right attorney
In general, history is the best indicator of future performance. Obviously this isn't the case (if it were pardon requests would never be considered), but you should find an attorney who as at least had some success with getting ex-offenders pardoned. Ask potential defense attorneys about specific cases where they have actually seen pardons granted.
Don't be surprised if many (if not most) of the attorneys you talk to have never actually seen a pardon request approved - pardons are very, very rare. In fact, a lot of attorneys might tell you not even to bother with a submitting one, but if you feel like your situation justifies at least consideration from the executive office, you should be relentless in pursuit of the right person to represent you.
When are pardons usually granted?
Nobody will openly admit it, but usually felony convictions are pardoned for political reasons. If, for instance, you were involved in a controversial case that may help a future candidate in a bid for a position, you need to play that angle up to even have a chance of a pardon. It helps if you are a high profile person - either knowing the right people or being a whistleblower in a popular case will significantly increase your chances.
Alternative to pardon
If you can't get a full pardon, you may still be a candidate for expungement. Sometimes you crime is not expungable but the court will downgrade the crime from a felony to a misdemeanor. Again, talking to a good attorney will probably give you a better idea of realistic options in your situation - make sure that you are completely honest with your attorney about all of the details of your case - they can't help you unless they know all of the facts.