Kill Chain occupies the Perk 2 slot, the same slot as perks like Ghost and Hardline, making it an expensive compromise. However, when used effectively, Kill Chain can virtually guarantee high elimination sequences. Normally, kills you get from killstreaks like Cluster Strike or Precision Airstrike don't count towards earning higher killstreaks. With the perk equipped, those kills will start counting toward killstreaks, so if you earn a decent killstreak that doesn't require too many kills, it can lead to something even bigger.
When equipping Kill Chain, you should probably re-evaluate what killstreaks you currently have equipped. The kill-streaks that last a long time should be a priority. A Wheelson only requires 7 kills and can easily double your kill count if you play your cards correctly. Another great choice is the VTOL jet. It requires 8 kills, so you won't get it too often, but when you do, you can easily head to a Juggernaut if you can stay alive while the VTOL does its job. If you manage to get a 10-kill streak and nab a Chopper Gunner, then you're pretty much guaranteed to get a Juggernaut before the streak ends. It gets harder to maintain a kill streak the longer you stay alive, but Kill Chain can make those final killings a cinch.
Of course, Kill Chain has some restrictions. First, you can't steal a teammate's heal package and use that killstreak. It won't count. Second, killstreak victims do not count as a tactical nuclear weapon. A nuclear bomb requires 30 kills in a row, but those kills must be done with weapons, not kill streaks. Still, the power of Kill Chain is well worth those stipulations, and you'll gain rarer Modern Warfare kill streaks much more often.