Both Recount and Skada allow players to look at both DPS and damage. Within the latter readout, it's possible to enter into the statistics for damage done, and get more valuable information on what you're doing your deeps with. One thing I really like to do when I'm remastering a spec after an expansion is to go into the Raid Finder and hope that I come across at least one other player who's playing the same spec as me.
If I outperform them, or if they outperform me, I have a look at the damage stats. Is their biggest damage-dealing spell totally different to mine? Where's their damage coming from? Is there a spell in their rotation that's doing far more damage than it is in my rotation? If so, then perhaps I'm not using it enough. Are their DoT uptimes higher than mine? Maybe they're lower? Is this causing this player to out-DPS me?
All this information is available in the meters, in Skada, in Recount, and no doubt in others, too. You can find basic information here without ever having to alt-tab out of the game, but for more in-depth monitoring, there are superior resources available.
2. World of Logcraft
What does this get you that you don't get from the meters, as mentioned above? Detail. World of Logs parses are far more detailed than the information neatly contained in those addons will ever be. You can view the number of hits, crits, misses, ticks, you name it, it's all there. You can compare your output with that of others through the course of the fight, seeing where your output went up and down, see when heroism was cast, all that sort of information.
Just like Recount and Skada above, you can use World of Logs to check out others' spells and abilities. For example, say you want to have a peek at what a top shadow priest is putting out? Well, head over to the fight you're particularly curious about by hovering over the Rankings menu and picking the relevant raid from the drop-down menu. Nothing but mages in the top spots? Fear not, click "more" at the bottom right-hand side of the boss's pane. This transports you to a page where, by using a series of drop-down menus, you can select exactly what you want to see. Select a shadow priest whose abilities are in a language you can read, then head to their player output. Then you can compare your data to theirs!
3. Talents, Glyphs, Rotation and Priorities
Rotations have largely gone the way of the dodo, but given the information you will have obtained from the exercises described above, particularly comparing others to your characters, you may have some idea of where you could tweak or improve your spell priorities. Priorities have largely replaced rotations, meaning that DPS is now rather like a flow chart, so if ability one is on cooldown, use ability two. If you have greater than quantity X of item Y, use ability Z. That sort of thing. And, fortunately, many people have worked out these priorities for you.
This information usually comes with input on talents and glyphs, as well as gearing, gemming, and enchanting. While the former, and to an extent the latter, are more down to player choice since the arrival of 5.0.4, the theorycrafters are always working on which offers better damage, better regeneration, better utility, and so on. The best places to find this sort of information are sites like Elitist Jerks, Icy Veins, Noxxic, and of course our own WoW Insider class guides. There are also other sites that cater to theorycrafting for specific classes, such as HowToPriest or Totemspot. Please do let us know in the comments about other sites you use!
For the ambitious, there's also Simulationcraft, which provides fabulously in-depth and detailed information, but is also very complex, and occasionally hard to understand.
4. Reforging and Robots
Once you've got an idea of your damage-dealing abilities and rotation, take a look at your gear. Armed with the information you've gathered from the aforementioned sites, you should be able to take an informed look at resources such as Ask Mr Robot. WoW Insider featured a great guide to getting started with Mr Robot a while back, which is very much still applicable. If you like Mr Robot, you may want to consider addons such as Reforgelite and Reforgenator, which, like Mr Robot, will do the math for you when working out what to reforge to get to the hit cap, haste breakpoints and the like.
These addons often come pre-loaded with stat weights and priorities, but will also allow you to create custom weights, caps and the like using data you've gathered from the aforementioned sites. Don't just assume that the preset stat weights and priorities are correct, use the sites to make sure you're reforging for the best stats for your style.
Ask Mr Robot will also suggest upgrades for you in gear, enchants, gems and so on, so is very useful indeed as a gear list substitute, but the aforementioned sites may also have other opinions. Use your new-found knowledge, don't just follow one site's opinion blindly, shop around.
5. Practice, practice, practice
One big secret to really good DPS is knowing your spec so well that using the right spell at the right moment almost comes naturally. That way, when you need to move out of fire or switch targets, or anything along those lines, you can carry on with your DPS. It's almost innate. And that innateness comes with practice. It's often referred to in the PvP community as "muscle memory", seeing as most players advocate keybinding and mouse movement over clicking, but if you prefer to click your spells, you can still learn your DPS priority list just as well. What's good practice? Well, target dummies for one. They're also a great place to test out your new abilites, reforging, gemming, talents, glyphs, all the information you've managed to gather from the resources above. See if you're getting a DPS increase from your hard work, or not.
And when you're out doing your dailies, or doing heroics, doing Raid Finder, or anything along those lines, don't just do what's needed. Use your knowledge, and try to do the best you can. Sure, quest mobs will likely die before you can get everything cast, but stick at it anyway. It's kind of satisfying.
What are your top tips for improving your DPS, regardless of spec?