People like table-top role playing games, but the general consensus is that a lot of the role playing has subtly turned more into combat roles than character roles. The main reason for this, I am going to posit, is pure economics. Resources, specifically.
The first resource is Time. The average time people have to game drops as they age and responsibilities kick in. The two times in your life when you have the most time to play games are at the bookends of it. For everyone else, there isn't time for a day-long delving into the lair of the dark lich. Combine this with the second resource/problem: Players, like any other actor in an economy, are rarely rational actors. These are the two biggest factors standing in the way of getting more RP in your RPG.
So, how do you get more out of your RPG?
Like any economic problem, you simply have to use your resources better.
I'll admit to something that is true for almost every player, and which is true for me as well. A player wants to accomplish something. Exploring the rich and varied world of the GM is, often, not high on that list of things they want to accomplish. Locked doors must be opened, plots resolved, monsters killed (or tamed for using in the killing of other monsters,) etc. Yet, as players, we're often inefficient at approaching these problems. Do you know how many hours of my gaming life have been wasted by someone saying: "Wait, let me add up my bonuses" or "We search the door. Wait, first, we search the square in front of the door. With a five-foot pole, unless someone has a ten-foot pole." Players waste Time.
Simplifying combat is one way to net more time for role-playing; having the players have a plan is one way to do it as well. A gentleman's agreement between the GM and the players can help as well: Think of how much time could be saved if every player did not have to clarify: "I eat my rations, unless they are obviously spoiled or appear poisoned. I check my rations for poison. First, though, I check the bag I carry them in for contact poison." Most every table top game in existence has a passive perception check: You don't get more chances to notice something by being paranoid. Stop wasting Time, on both sides of the screen, and play the game.
The next problem with Time is, like most resources, the equitable distribution of it. There is one GM for up to four PCs. If the players aren't role-playing with each other, the GM is going to be hard pressed to keep them enthralled in the action. You could run with fewer players (or more GMs), but either way, the goal is to ensure more people are using that same parcel of time at once. This is why combat draws to a standstill in most games: Everyone isn't engaged (or, worse, everyone is telling the wizard what to cast where), so you lose 10-15 seconds each change of turn as Players look away from their handheld devices back to the screen. Then ask: "How badly hurt is that guy? Is anyone worse hurt? What about us? Who is hurt?" Paying attention saves Time, efficient miniatures will save time too. Get markers to show how badly hurt something is; give players cards to show how many hit points they still have out of their total so people can eyeball the situation. Anything that increases the player's situational awareness is a good thing.
Finally, Players often come to the game with a different idea of how they will spend their Time. For some players, witty banter with the quest giver is enough role-playing. Other people don't have their fill until their bard has drafted an impromptu sonnet about the tragedy that has befallen the inn keeper's daughter and the quest that is to be undertaken to save her. Needless to say, finding compatible players is a lot like finding investors with the same tolerance for risk, where in this case, risk means sonnets.
The Players also have a different idea of how the Time should be spent than the GM. The GM is running the time share, so theoretically, the GM should get final say. But you'll have a lot more Time if you run out of Players, so again, you need to calibrate to what you're both happy with. Which is ultimately, I guess, the key: Play the game, have fun.