Ask the GM
I went through several articles in the Bullet Points column archive, but found none mentioning the the smart hero's talent exploit weakness. Little has also been said by fans about the talent. This leaves how the talent works in the hands of the GM. Below, then, are two possible readings (therefore allowing the GM and the player to reach a compromise between the two) and a frame challenge.
A conservative reading
Anyway, at its most restrictive, the smart hero at the beginning of the second round of combat only either picks one opponent to be the subject of this talent (this is not an action) or doesn't use this talent during this combat encounter. If the smart hero opts for the former, the smart hero, only on his turn during the second round of combat, can take a move action to make an Intelligence ability check (DC 15). Failure means no benefit during this combat encounter. Success means for the combat encounter's remainder the smart hero uses his Intelligence modifier to determine his modifier on attack rolls against the subject instead of using his Strength modifier or Dexterity modifier for attack rolls against the subject.
(It's possible that the talent mixes fluff and mechanics for designating an opponent. That is, After 1 round of combat, the Smart hero can designate one opponent may be tied to the next sentence rather than to and try to find ways to gain an advantage by using brains over brawn. This would make it so that the smart hero need not pick the subject at the top of the round and could, instead, pick the opponent at the beginning of his turn. This is also something to work out with the GM.)
A liberal reading
At its most generous, the talent exploit weakness allows the smart hero at the beginning of the second round of combat and every round thereafter to designate an opponent. Then, at any time during that combat, the smart hero can take a move action to make an Intelligence ability check (DC 15) versus any designated opponent. Failure means the smart hero may later take another move action to repeat this process to try again versus that designated opponent. Success means for the combat encounter's remainder the smart hero uses his Intelligence modifier to determine his modifier on attack rolls against the subject instead of using his Strength modifier or Dexterity modifier for attack rolls against that designated opponent.
Further, under this generous reading, the talent's effect continues no matter how many opponents have been designated or have had their weaknesses exploited. That is, like Dale's answer explains, the smart hero gets a metaphorical cookie each round after the first.
This second reading requires the smart hero's player to track separately both the foes that have been designated and if those foes' weaknesses have been exploited. Because it's more complicated than only tracking whether the smart hero has exploited the weakness of a lone designated foe, this second reading is unlikely to have much traction with the GM, but, as many d20 Modern special abilities are at least disappointing if not outright disempowering, scrounging for any edge—no matter how small—might be worthwhile.
Challenging the frame
Official sources included, my research shows that few folks care enough about the talent exploit weakness to mention it. The most poignant fan assessment of the talent I found asks What d20 Modern character does not have Dexterity as his second highest ability score? That is, there are likely better talents to take than this one, which—no matter how the talent's read—allows a character, after the first (and most important) round of combat to sacrifice a move action to make a nontrivial ability score check and, if successful, gain against a specific lone foe a +1 to +3 bonus on attack rolls. In other words, to challenge the frame a bit, does the character already have the talent plan?
(Also, consider investigating Back to Basics, a conversion of sorts of d20 Modern's classes to Spycraft 2.0… for inspiration if nothing else. While I've a playtest credit on this supplement, I receive no other compensation for my shilling. I just want more folks playing Spycraft 2.0.)