In my experience with Pathfinder, there are two surefire ways to encourage people to interact with each other in role play: have each player make a game-relevant background and throw problems at them that have no short-term answer
The first and easiest way to get people to interact in game more is to have them follow this background guide as the first step in creating their characters. When most people who aren't familiar with table-top games create their first character, their first instinct is to fall back on Video Game RPG logic and construct a PC that appeals to their wants rather than their imagination. For example, a guy that is into something like Harry Potter could be thinking "Man, I'd love to be a Hogwarts Wizard. HEY, there is a wizard class. WITH A TON OF MAGIC. AWESOME, I SHOULD BE ONE!". And with that line of thinking, he either makes a perfect bad ass in every way mystically (that tries to be a glory hog by doing EVERYTHING with magic) or some mopey little twat that sits on the side-lines (because the person playing them is miffed at only having 7 spells a day). However, by having them start with the background sheet (with one or two subjective re-rolls), they get a view of everything that their character came from rather than what they can be right off the bat. For example, if someone had the misfortune of rolling Outlaw Profession Con Artist, the guy could be like "I could deal with that", and then skim through the classes thinking of how to make the profession/background work in their favor. He could be lazy and take the rogue for simplicity sake, but then he stops at the cleric thinking "Hey, wouldn't it be funny if this priest goes around swindling people while preaching the good word?".
And now you have him hooked. From there, he isn't looking for what feats or powers he would like to buff his PC with to make him Tru Soopor Sand, he is looking for what feats and powers work with his idea. Eventually, The good sister Loralia of the Elvish Isles walks into a tavern, who proclaims the admittedly unfocused teachings of Seraph the Sunbather. As she picks the pockets of the old geezer in the corner, a glint catches her eye. A blade that sparkled like a shooting star is a scant five paces away. "Surely such a magnificent blade could cut through stone like butter!" she thinks. Looking just a little higher cleanses her palate and throws her into disgust. The shooting star is on the belt of some ugly half-orc beanpole that was too small to fit his bearskin jacket. Steeling herself for the opportunity to take the blade for her own, she strides up to the runt, swaying like an ocean. She puts her hand on his thigh and asks "What is your name, my child?". Two things happen. First, he stammers with a blush that he is Roc, son of Kornak, who seeks to do his father proud by claiming many heads with the familiy Longsword. Second, Loralia realizes too late that the sword is secured too firmly to Rocs' belt to pull away. From there, the guy could get his character in a bitter fued with Roc, or start a long quest of Romance and Betrayal to take the sword. All of which is happening in the background of the main campaign!
The more complicated, and equally effective way to get people more involved in your campaign is through long-term dungeons. When I suggest this, I don't mean something as long and arduous as spelunking from one side of the continent to the other and kill fifteen dragons on the way, I mean present an overarching problem and drop them into the middle of it. For example, in a previous campaign that I played, our overall quest was to travel across the countries of Flails' Head and gather the remaining Hero Kings to destroy the one that turned traitor, killed an other Hero King, and declared war on the whole continent using his hordes of the undead. Our first stop on our quest was this little bo-dunk town where our contact was supposedly hiding in; we found a ruin that was buried under the town instead. After clearing the dungeon, we were appraised by its watcher, who turned out to be this crotchety old lady that was at the inn when we found out about the abandoned fortress beneath the town square. She informed us that the King, who we were led to believe had taken a leave of absence to wander the world, had be usurped by some upstart spell-caster that threatened to level the capital if he didn't stand down. After leaving the city, he was captured and had his soul rent into four chunks that were hidden or entrusted to her commanding officers.
Other than 'You have to go to the city to revive the King' and 'I will meet you here, but get lodging elsewhere' that was all we got. We got dropped off at the main gate, got let inside, and the rest was up to us. By making us find our own allies and do our own planning, our DM put us in a position where we had to actually work as a team while he worked off of our ideas and set up encounters that either challenged or completely wrecked our plans. For instance, there was a part where the cleric was wandering around and just happened to find the Dock Masters' lost pet duck. After we returned the pet, he idly mentioned about the constant threat of sabotage he and his guild had to deal with. We offered our services and set up about twelve traps around the boat to alert us when the saboteurs were among us. At around midnight, we found a group of street thugs dressed in all black that proudly claimed that they were going to blow up the ship when we sprang out of the shadows. However, we didn't realize how obvious moving all the tables and shipping crates to surround the boat was. Moreover, we didn't think for a second that these guys were a distraction that their Dark Paladin leader used to set the bombs on the opposite side of the ship. That was a fun night.
TLDR: Instructing your players to roll for their background first when they make their character creates a strong connection to them, one that makes them feel like playing the roll of the character rather than the class. Putting the party in a long-lasting dungeon that gives them no other direction than "Save the Day" promotes teamwork and a good situation to play out their characters.