I've been doing quite a bit of research on this topic (reading FAQs, watching replays, and consulting with others) in the last few weeks, and I think I can put together a pretty good strategic overview at this point.
Know Your Role
In CoD multiplayer games, you are generally doing one of three things:
Defending - You have an established position and you are actively attempting to hold it. In some game modes this is actually the point (ie, Domination) and in others this might be considered "camping." Camping has some negative connotations in first person shooters, but in CoD it's a perfectly valid and encouraged strategy. Determining the right position to defend, and the manner in which to defend it is map dependent, but having several friends with which to divide up the responsibilities can make even the most vulnerable position almost impenetrable.
Defenders generally need to focus on close range weaponry and anti-personnel equipment. Shotguns, SMGs, mines, and close range perks such as Steady Aim are all important. It is possible, however, to be defending players who are entrenched in sniper positions who are focusing on long-range kills. In these cases, it's best to ensure you have struck a good balance between close and long range firepower in your team/group's classes.
Assaulting - You are aware of enemy positions, and you're attempting to overtake them. The assault player needs to focus on disrupting enemy defenses with tactical grenades and indirect fire weapons such as the RPG. Bullet penetration is also important.
Playing assault can be a great benefit to your team, by allowing them to break up enemy sniper nests and disrupting player killstreaks. However, having assault players running into a heavily defended area one at a time is the defender's best case scenario - they are ready for you and they need your kills to earn points. Be aware that sometimes the best idea is to avoid areas that are heavily defended, especially in non-objective games. In objective games, consider an alternate objective point if the one you are assaulting is heavily defended - there are only so many people on the enemy team, and they are likely to be focusing on fewer points.
Advancing - When you are unaware of the enemy's location, and you are actively seeking them out, you are advancing. Advancing players may be trying to get to an assaulting or defending location, or they may be playing a "run and gun" class where they hope to encounter players who they can quickly and/or quietly dispatch.
When advancing, pay attention to corners and lines of sight. Tactical grenades and indirect fire (lethal grenades, grenade launchers, etc) can help flush out other advancing players and defenders.
Two groups of advancing players frequently clash with one another, and the result can be what I refer to as "civil war" - both teams are assaulting a location with limited defensive capabilities, dying, respawning, and running back to rejoin the fight. In this situation, it's important to realize that spawn points can shift, and players can frequently flank. Be ready for a surprise coming from a direction you're not expecting, and look for alternate routes to the current area that you can use to surprise your enemies.
Know Your Class
There are, in general, 3 types of classes in CoD.
Close Quarters - Close quarters weapons include SMGs, shotguns, pistols, and the like. They are lethal at close range, although their damage output quickly falls over distance and they do little to nothing when you are far from an enemy. In addition to damage falloff, they tend to have serious accuracy issues at range.
The close quarters player needs to move fast, melee often, and may prefer to hip-fire or sneak to get close before engaging. Avoid open areas or long lines of sight. Play the "corners game" - you always want to be close enough to a corner that you can be at close range to whoever is in front of you, and close enough to duck around it in case a firefight erupts behind you.
Balanced - Balanced weapons include primarily assault rifles, although some light machine guns can also fit into this category. They can fill almost any role, although at extreme close range they are at a disadvantage to fast moving close quarters players.
The Balanced player can pick a variety of perks to accentuate their play style. For instance, add a silencer or a heartbeat sensor and some perks to counter enemy UAVs, and you've got a fairly mobile short-range flanking class. Add an ACOG scope to a rifle with low kick, and you can provide deadly long-range cover fire.
The balanced player will want to keep close range players away, while making sure to avoid extremely open areas, although they may be somewhat more evenly matched against a distant foe.
Long Range - All sniper rifles, certain assault rifles, and many LMGs fall into this class. The long range player is generally easy to blindside, but can take out foes at great distance before they have a chance to become a threat. Thermal scopes, point defenses (claymores, etc) and a strong secondary weapon are important to this class.
The long range player needs supporting players more than any other type, in order to protect against close threats and to give them time to line up shots. They can support other players who are engaged in direct firefights by doing heavy damage to and finishing off players who might be out of the line of sight of others. Knowing the proper positioning of point defenses and the optimal lines of sight are important to the long range player.
Heat is an important topic when discussing CoD multiplayer. "Heat" is a term I use to describe what parts of a map see action. Managing heat is an important part of playing the game. As a defender, you want the area you are defending in to be somewhat hot - enough that you're getting kills, but not enough that the enemies are overwhelming. When you are moving behind enemy lines, you need your heat to be low - using silencers and perks that keep you off enemy radar will help cement the element of surprise.
Some areas of a map are naturally hot, and some naturally cold. However, using unsilenced weapons and making many kills in the same location can increase the heat. If defending, you can switch up your positions in order to avoid the effects of overheating a particular location.
Lag and reflexes play an important role, although by and large they are beyond your control. Dealing with these issues can be frustrating, but it's important to keep a cool head, and be thinking about your tactics. If someone is killing you in a location repeatedly, you may be tempted to continue attempting to break their defenses by running into the same area over and over again. However, because of the way killstreaks work, this is just feeding the enemy team and making things harder on your team. If an area is home to an enemy you just can't seem to defeat, make that area cold. Keep away from it and encourage your teammates to do the same. This will force the enemies to change their tactics in order to get kills. These new tactics might be easier for your team to break.
Balance is a touchy subject in CoD games. Usually, there are weapons and perks that are clearly superior in a given situation. Know your class' strengths and weaknesses, and avoid areas where you would be at a disadvantage. For instance, a class that is heavily close quarters should not run out into large areas or down long halls. Experiment with different weapons in a given class to find the ones that are superior, and mold your classes around these weapons and equipment.
Remember that death is cheap in CoD games. Don't be afraid to consume your loadout, because it is likely that you will respawn many times. Nothing is worse than dying because you didn't utilize your tactical grenades. On the flip side, nothing's more satisfying than laying a random claymore that gets you a free kill.
Know the maps whenever possible. It's a good idea to load up a private match with a friend and practice various grenade throws and approaches to common camp spots. You can even do this in split screen or with bots - move someone into position, and then practice various angles of attack to get maximum benefit.