Refer to the source
If you want to create a campaign close to Buffy feel, you must watch or re watch Buffy, paying attention to get ideas for the game you want, and how they implement them. You must understand why they make the things they make in the way they make.
In my opinion, Buffy as a (comedic) supernatural teen drama has the best balance between normal life and supernatural plots of its genre (that I have seen), at least it's the most entertaining.
Ally with your players
You need them to create motivated characters, with interests on family, friends and romance relationships, as well as other mundane goals, such as getting good (or sufficient) grades, be popular, get money, look cool, ... Characters also need detailed personality and merits and more importantly flaws (as being a bad student, socially awkward, jelous,..).
You will also need them to be proactive to acquire mundane objectives (for example getting a date with someone), and to share them with you, possibly with the group. This way you can prepare the plot and related scenes. Bonus if they also suggest problem for their characters (as someone trying to compete in romance, or a familiar getting sick).
Finally, players must know that relationship their characters have with other player characters: friends, allies, romantic interest, friendly rivalry, ... (I recommend not to make them enemies), and to interact between them in that terms (In Buffy protagonist talk a lot between them and share their points of views). They should try to keep their character somewhat tied to the group, and not to make others away from them. If two PCs get angry at each other, they should finally find a way to solve it in the medium term, or one PC must leave the "party" and a new one must be created.
Sketch the season
I could write quite a lot about how to make Buffy-style seasons, but I think the questions demands for much generic advice.
You should lists the plots you have (mundane and supernaturals) and create related events for them. Then distribute them between episodes (game sessions).
Of course, as you are playing an interactive form of fiction, those sketches must change continuously, reflecting you players actions and interests. But an updated list help you don't losing any plot and to prepare a good mix.
Prepare the scenes in advance
As said before, watch Buffy series to know how to get the feeling. Look at each scene and think why it is added. A scene should always have at least one plot objective. You don't see every meal, every class or every conversation. You see the scenes that matter for something.
Face every session as an episode, and prepare scenes related to the plots your characters have at this point (such as an NPC asking a PC to go out, or a teacher picking on a PC, or a PC mother having a new love). Also, be prepared to create scenes for what your players want to do (the more you can advance, the better), and try them to be interesting and meaningful.
When playing those scenes, try to be short. Many of them don't concern every player, so all the others are waiting. If the scene is interesting, players won't get bored, as if they were watchers of a TV series. But you must try to end them quickly, or they will lose the interest. This can be harder that it sounds.
Making the mix
The hard thing here (and on a TV series for that matter) is to get the right mix of supernatural and mundane attention. This must be thought for each game session, as well as for the whole season. The amount and the pacing depend heavily on personal tastes, so I will tell how I would do it:
Supernatural plots are primary, they get like the 66% of the attention. They are usually more urgent that personal life (lives are in risk), although an evaluation test is also quite urgent.
Mundane plots are secondary, but they are also usually more long termed. So, today I must stop an evil cult incantation, but next week I will still see my love interest. It's like a background in motion.
The question about this balance is: are characters trying to live their lives while dealing with supernatural danger, or are they trying to face supernatural threats while dealing with their lives?
Make a good system choice
If you haven't already chosen your system, think of how every of them is going to impact on your campaign. Each system is tailored toward some objectives, so choosing one aligned to yours is a key to success.
For instance, in my opinion, D&D would be a terrible choice for the campaign you are talking about, but also BSR, Rolemaster, and many others.
I feel very comfortable with World of Darkness system, and I have used for similar games as yours. It has a physical/social/mental balance that I feel very appropriate to represent students and high-school situations. The background system is very useful, once you readjust it (Resources meaning how rich is you family, Fame meaning high-school popularity,...) and a key to a very social game. Of course, players need to create believable characters (for instance not having Firearms 3), but even that can vary according to the campaign tone (many Buffy characters would have ridiculously high ratings on some skills).
Of course, there are systems that are more tailored to that specific situation. In other answers, you can read some recommendations like Monter Hearts. I have not played any of them, so I cannot give you advice.
Just try to know the system you are choosing and think about the game and plot impact of the main mechanics before making a decision.
Best Answer
Solution 1: Harlequin
Have an extra player play NPC's. At start of scene, give them a card with their goals for the scene, and the NPC's sheet. If needed, give them also some information about what that character knows about the bigger picture. (But note that what they know may or may not be true.)
Solution 2: Expand the structure to 2+ Teams
Split the focus between team 1 and team 2, with players on each team. For example, if doing, say, any given police drama, each episode focuses on one team as the big, A-Story characters, with a smaller B-story by the other team. For Example, in Law and Order SVU, Bensen and Stabler are team A, and Tautuola and Munch are team B - in any given story, one of them is the lead story, getting about 2/3 of the episode, and the other team gets about 1/4 the episode. (minor overlap covers the remaining 1/12).
Solution 3: Narrow the Focus of the Characters
In Classic Trek, the "Trio" is Kirk, Spock and McCoy. Kirk is the action man and romance man, Spock the brains man and logic man, McCoy is the Heart Guy and medic. My last 4-player focused Bridge-Crew Trek game had the Captain who was Brains Man, psionics man and Social Sciences, the First Officer who was the Heart Guy and Action Man, the Doctor, who was both medic and logic guy, and the security chief and psionics guy. Note that the romance guy is absent, and the additional role still leaves some empty space for a 5th player. (Or that empty space can be used instead for some comedy of errors.)
Solution 4: Main or 2+ supports
Have players choose either to play 2-3 support characters, or one main character. Those who pick support characters get reduced points, and their two or three characters should be fairly different.
For example, if doing L&O Criminal Intent, Player 1 is Goran, Player 2 is Eams, Player 3 is the Captain and the Shrink, plus maybe Eams' boyfriend (in the show, not seen). A fourth player could be the Coroner and the ADA, and possibly Goran's mother.
TV in General: The Relationship Map.
The most important element in good TV emulation is a good clear map of the relationships.
Take, for example, Buffy. Who are the main characters? Well, for the series overall, Buffy, Giles and Xander. Why? Those three are in all seasons, and all other characters hang onto one or another of them. Monsters mostly attach to Buffy or Giles. In any given season, 1-3 others are 'main cast' - but the others vary by season. Anya, Tara, and Dawn are later seasons only. Willow goes away for most of a season, as does Giles. Giles doesn't get written out, tho - he guests every couple episodes - and his presence is still strongly felt even in his absence, and his prior relationships drive both a couple villains and the second tier characters. Willow's absence was fan-hurt, but not so much story loss, as most of her relationships overlap with Xander and/or Buffy.
Mapping the relationships out, however, reveals just how ensemble the plotting is for BTVS - Spike is linked to Angel, and Angel to both Xander (hostile) and Buffy (Romantic); Spike becomes linked to Giles, and later everyone. Anya starts linked to Cordelia, but then converts to Xander, and with Cordelia's absence, takes over Cordelia's role as Xander's love interest, and eventually Giles, as business partners.
Mapping out your party ahead of time allows them to come in with preestablished relationships, which makes the group dynamic much easier to maintain.
Round Robin with Troupe Style Rather than Group
Good start of game mapping can make it much easier to get a TV type feel. But also note that in TV-Land, not all the characters are in every scene. This can be emulated with what Marc Rein-Hagen dubbed "Troupe Style Play."
In Troupe Style play, every player has a main character, one that they alone play. The group also has a collection of secondary characters, which anyone not in the scene can play. These troupe-characters are included as needed, and usually, one or more are present in most scenes. They don't even have to be the same character to the same player. Focus on 1-2 of the main characters in the scene, and those not involved get to play a troupe character who could be in the scene.
Borrowing from Law & Order again, Dale Stucky could be a troup character, as could Dr. Wong, and any of the ADAs, plus a handful of recurrent uniforms, the Tech guy, and the Coroner. In such a game, Tautuola, Munch, Stabler and Bensen are all PC's. Only 1 or 2 scenes per "episode" would be all 4 PC's; all the others might be 2 pc's and 1-2 extras. Or 1 PC and 2-3 extras.