The question is really in the title. (oops.) Is there some kind of special assignment I have to do to rank up my specialization, or do I continue to rank up normally? (I picked Stalker, and I don't know if I should continue leveling up like normal, or if there's a specific task I have to complete in order to get the next Stalker "rank"... help?
Leveling up follows the same process after level 50, with the small exception that you have to choose a Specialization. If you do not, you will stop gaining XP (as I found out the hard way). Once you have chosen the Specialization (as in your case), you just keep doing what you did to get to level 50. The only way to gain XP is by playing - and primarily you gain XP by playing War Games and Spartan Ops missions (with the latter being significantly advantaged in the XP department).
One other slight nuance is that Specializations unlock a Support Upgrade or Tactical Package once you get to the 10th level of the Specialization. Other than the cosmetic benefits of the armor, weapon skins, emblems, etc - the level 10 unlock is the main reason for each Specialization. Just because you unlock it at the 10th level does not mean it is automatically activated - you need to go to the Loadout Customization interface and choose the Support Upgrade or Tactical Package at that point.
More information than you were probably after, but I thought I'd share.