The first chunk of the COD MW2 mission 'Alone' doesn't include a gun or gives the option to equip one thus we skipped ahead to the fountain.
As you approach the fountain part, make sure to grab bottles, tape, chemicles and other supplies. Continue to craft various equipment by accessing the backpack and don't be afraid to use your knife.
Both the throwing knife and your melee knife will help you and NOT void the achievement. The same goes with the proximity mines. USE YOUR EQUIPMENT. There are parts, for example, after you leave the bookstore after the fountain that require you to just run despite being shot at hence why it's almost needed to try the MW2 Gunless Achievement on Recruit difficulty.