Render unto Caesar what is Caesar's
You were not responsible for hiring this person, and you are not responsible for firing him either. Any action you take should be based on the premise that the ultimate responsibility for this person lies with the people higher in the food chain who are paid to make these decisions.
Render unto your Boss what he needs to Lead
At the same time, it's not your job to protect a coworker's folly from your boss. If he is really this bad, then your boss should already know. If not, how can you expect your boss to make the decision?
You say, "I don't want to be a narc because I don't want to sour my other coworkers to me", yet if he is as bad as you say, don't you think they are already uncomfortable with him as well?
Fair and Balanced
To prevent this causing problems, you need to be fair and balanced in how you approach any claims against your coworker both with him and your boss. I suggest a two-prong approach.
Communicate with your Coworker
First, you need to explain to your coworker that his level of work could use improvement. This does not mean criticism, it means giving constructive advice on how to improve. Rather than getting angry for missing deadlines, why don't you just ask him how he manages his task scheduling, and explain your process as well to exchange thoughts/ideas? Of course your actual goal is to bring to his attention the fact that:
- You realize he is missing deadlines
- This is something that can be improved
If he ignores it, he ignores it, and you at least have given it the good ol' college try. A similar approach to his other errors can work too -- try to make it friendly, not managerial (you are not his boss), and at least try to help him improve himself before moving on to the second part of the approach.
Communicate with your Boss
If you manage to fix your coworkers' poor habits 100% this isn't necessary, but most of the time people are not able to change overnight. So even if your coworker is improving (but still making careless mistakes), you want to talk to your boss to protect yourself long term.
Since it is your boss' decision, you become a part of the problem if you hide information necessary to make that decision.
This does not mean complain about the mistakes. It means bringing up facts in an unbiased manner to provide guidance to your boss on what the situation is, and letting him/her decide how to handle it. Something like this:
Hey boss, as you know I've been working with Coworker. I was hoping you could find some time to work with him and help bring him up to speed with the rest of us. I've done my best to train him on X, Y, and Z, but he is still making some rookie mistakes like missing deadlines and sending out e-mails with the client's name spelled incorrectly. I'll continue to do my best to teach him what I know, but I think it would be best if you stepped in to help out.
You are just stating facts (things you should be able to back up instantly, especially deadlines and e-mails), and you are not making a judgment. You are stating you are doing your best to bring him up to speed already, and are saying you will continue to try (both should be true!).
Your Boss' Decision is Final
At the end of the day, respect your boss' decision. It is his/her decision to make. If your boss decides to have you continue to do your best to teach them, then respect that decision and ask how you should report progress to them (or more specifically, the lack of). Questioning their judgment or pushing for a specific result will have a much bigger chance of poisoning the well.