Some system agnostic advice.
First, remember that roleplaying happens (or should) at all phases of the game - it isn't just something that happens outside of combat and it should IMHO be something different than just skill checks. Opponents, even in combat, can talk to and taunt the party and if you set the right tone the players should talk in character even during combat.
Second, just because one character is the "diplomat" doesn't mean that NPC's have to address that character to the exclusion of the rest of the party. Don't be shy about engaging every character into interactions with NPC's.
Third, as others have mentioned getting your players to write up backgrounds for their players gives you (and them) hooks into those characters and helps you as the person running the game find hooks to engage each character (and equally important) each player into the roleplaying aspects of the game.
If it helps make sure that you rewarding the players for roleplaying interactions (do this mechanically if you need to) so that players don't feel like they are "wasting time" if roleplaying interactions keep their characters from "advancing".
That said, one of my favorite gaming memories was an old campaign I ran in high school (over 20 years ago to date myself) where my players (and I) were so engaged in roleplaying we went many sessions between rolling a single die and while the players didn't gain many levels over the course of the campaign they had a lot of fun - and at the end of the day that fun is why we play games.
One other "trick" I've used in the past is to create situations where there are many routes to success for the players - this includes situations which might involve combat. As your players start to realize that combat isn't their only (or even their best) option this can often help all of the players get into roleplaying (don't just let them get away with a few rolls of the die to get out of a combat situation).
For that matter it is often good to force players to describe what they are doing - the effectiveness of which may be determined by a roll of a die - but make sure that they have to describe their character and what that character is doing to cause the roll of the die.
One other tip - ask players to describe their characters at other times - such as in combat. Over time forcing the players to describe their characters - physically and "in action" helps everyone at the table (including that player) get a more nuanced sense of each character - this then greatly helps foster roleplaying between the players (and between the players and NPCs).
In terms of balancing the party this starts at character creation but it also is reflected in how you run the game - like a good host at a party (in the "real" world) you should make a point of engaging with everyone at the table - don't let anyone person (including yourself) dominate the conversation.
On tip - do character creation a joint, group exercise - make it part of the game experience and help the players come together as a party as they are making the characters. This can be tricky (there are some game systems that directly suggest this) and if you don't meet often it does take precious game time - but the effect is that every player knows a bit about each other player's characters and as a group everyone has a sense of why their characters are together as a party.
Above all get players to show what they are doing - to describe it - whether "it" is a new power in combat, how they try to track down clues or a contact in town or how they go shopping (in character).