To unlock the "Spider-Sensible" trophy in Marvel's Spider-Man, you will need to perfect the Perfect Dodge move which you can find in the Skill Tree.
When you first upgrade your skills in the game (shortly after your first boss fight), I would recommend unlocking at least the first skill in each tree. The first skill in the Defender tree (centre) is the Perfect Dodge, which allows you to dodge an opponent's attack at just the right moment, automatically hitting them in the face with a web shot and generating some Focus. You'll know if you did it right if time slows slightly when you complete the dodge.
The key to Perfect Dodging (which is pretty crucial in effective combat anyway) is to keep your focus on Spidey's head instead of enemy movements. You need to dodge when the Spider-Sense glows, not when you see an enemy make a move.
For a Perfect Dodge, you need to wait until the last moment, when the Spider-Sense halo around your head briefly glows blue. Tap
at this point to execute the Perfect Dodge.
If you are trying to earn any XP in random crime encounters, Perfect Dodges will often show up in the additional challenges for the encounter (Press [cn_R3] to see those). Therefore it won't take long for you to unlock the trophy through natural progression -- in fact, it'll probably be one of the first non-story trophies you acquire.