This post is probably coming at a strange time, considering that the Cataclysm and with it the next gear reset aren't far off. Many people have stopped playing, and of those that still do, many probably play less than before, not chasing the next gear upgrade with quite as much enthusiasm and playing their alts very casually. On the other hand, maybe you need to get an alt geared up more urgently than ever to fill that vacated tank spot in your floundering raid group, so hey, why not?
Step one: Set yourself a realistic goal.
While it is possible to get even a non-raiding alt kitted out in full tier ten, it's a slow and tedious process that involves doing the daily heroic literally for months. Efficiency is really something else. Even better, should you ever get into ICC after all, a lot of the drops there and the frost emblems that you earn will be useless to you since you can't upgrade any further. Just settle for tier nine and its equivalents. It's easy to get, looks shiny and will serve you just as well for the rest of this expansion.
Step two: Formulate a plan.
I can't help but frown every time I see someone who wears for example tier ten shoulders and a blue cloak, as it's a sign of a rather inefficient approach to gearing up. Don't just use your emblems to buy whichever item you can afford first because it's cheap or looks shiny. Personally, I always approach the issue of gearing a new alt like this:
1. Which gear slots can I cover with emblem of triumph gear?
Things you can buy with nothing but emblems of triumph should be your priority because they are both easy to get and of good quality. Since you can buy a full set of tier nine that way, you've got the head, shoulder, chest, hand and leg slots covered. For most classes there's something for the ranged slot too, as well as a spec-appropriate ring and at least one trinket - buyer beware though, some of the triumph trinkets are pretty rubbish. Don't replace a possibly superior heroic drop just because it has a lower item level.
For the head and shoulder slot, you'll also have the choice between your appropriate tier piece and a non-tier item with a slightly higher item level. If the four-piece set bonus of your tier nine is very good, make sure to get exactly four pieces and fill the spare slot on either head or shoulders with the non-set piece, picking whichever one you think is the bigger upgrade compared to the tier alternative. If your tier bonus is useless, you might as well get both non-tier items.
2. Which gear slots can I fill with drops from the ICC five-man heroics?
Forge of Souls, Pit of Saron and Halls of Reflection on heroic currently offer the best non-raid gear in the game and should allow you to fill most of your remaining gear slots with something nice. Good weapons, necklaces, rings, belts, boots and trinkets can be found here. It doesn't take long to quickly glance through the loot lists for those instances on Wowhead and see if there's anything you'd want for your alt.
3. Which gear slots can I fill with drops from the ICC five-mans on normal or Trial of the Champion?
The ICC heroics offer a lot, but not everything. For example you won't find any shields there, and personally I've found the bracers on offer to be lacking as well. Most holes like that can be plugged with only slightly inferior loot from the normal modes however. Again, just check the loot lists. Trial of the Champion on heroic offers the same item quality as the ICC normal modes, and normal Trial of the Champion is a great place to go to for trinkets that are much better than their item level suggests.
4. Which gear slots can I cover with older types of emblems?
If you're still sporting a blue in a certain slot with no upgrade in sight, check out what the vendors from previous tiers have on offer. Depending on your degree of desperation it might be worth trading down some currency once you've made all your triumph purchases, to buy an Ulduar-level cloak or bracers for example.
5. Which gear slots can I cover with emblem of frost gear?
Now, even if you're not making a point of farming for emblems of frost, you'll likely accumulate at least a couple, running the daily heroic a couple of times and from completing the attunement quest for the ICC five-mans. Spend these wisely, which generally means not using them for a slot in which you can get something of ilevel 232 quality or higher anyway (i.e. no tier ten shoulders just to look cool). Cloaks and trinkets are my personal favourites for this one, but depending on how you've been doing with drops, something else might be viable as well.
Step three: Don't run your heroics randomly.
Sure, one random heroic per day to get both triumph and frost emblems is fine, but don't fall into the trap of chaining random after random just to be able to buy your tier nine faster. Emblems of triumph drop from all heroics and even the ICC normal modes. You'll be raking them in faster than you can spend them in no time, and if you start by accumulating emblems and then run specific heroics for their drops, you'll end up spending way more time on the whole process than needed and acquire a whole bunch of extra emblems that you might not even have a use for.
In other words, run one random a day at max, then queue up specifically for the normal and heroic modes that drop items you want. If you only just hit eighty, normal Trial of the Champion is a good place to start as well. You won't be getting any emblems there, but you can work on getting a good trinket right away, and the other drops - even if you don't intend to keep them for long - make for a solid heroic starter set to help you pull your weight in later instances.
Question: What about bind-on-equip gear?
I didn't really cover BoE gear here since it's either vastly inferior to anything you'll soon acquire through emblems and instance runs anyway, or about on par with raid gear and very expensive. Either way I hardly ever consider it worth the effort for an alt. For you it can vary depending on the circumstances however - if you are a crafter and have sufficient materials lying around anyway, it might be worth crafting some starter gear for yourself, even if you'll replace it soon (especially if you're a tank). Or if you have more money than you know what to do with, sure, go ahead and spend it on getting ICC-quality BoE gear for your alt.
On average, just making sure that all your gear slots are filled with either a piece of triumph gear or an equally good instance drop works pretty well however. Just don't forget about those gems and enchants!