
BigNorseWolf |

You gain Improved Unarmed Strike as a bonus feat. You can deal lethal damage with your unarmed strikes, and the attack doesn’t count as archaic. At 3rd level, you gain a unique Weapon Specialization with your unarmed strikes, allowing you to add 1-1/2 times your character level to your damage rolls for your unarmed strikes (instead of just adding your character level, as usual). If you can already add 1-1/2 your character level to your unarmed attack rolls (such as due to the vesk natural attack racial trait), you instead can add the block, grapple, and stun weapon special properties to your unarmed strikes.
Is that third soldier level, or third level overall? Most class abilities go with class (edited) level, but most weapon specilizations just care about your overall level.

HammerJack |

From CRB page 59 we have
Class features may reference different types of levels, such as character level, class level, and caster level (if you’re a spellcaster). For a character with only one class, these are all the same thing. If you decide to take multiple classes, when a class feature has an effect or prerequisite based on your level, it always means your level in that class, not your total character level (which is the sum of all your different class levels). If a class feature or spell mentions your caster level, that refers to your combined levels in all spellcasting classes (though the number of spells you can cast per day and the number you know at each level are still based on just the level of the class granting them).
The effect of the special weapon specialization says "character level" which is clear. But since "at 3rd level you gain" does not specify, and therefore means 3rd Class Level is when you gain it.
That's actually the same as normal class-granted Weapon Specialization.