Make Story Delivery Unobtrusive
If you're creating a JRPG, chances are your story will be delivered through text. There are many techniques to make text less annoying. You should absolutely have an easy to find text speed option that allows the player to make text appear faster or even instantly - slow text crawls are painful and feel like a waste of time to fast readers. Write your dialogue snappy and compact - avoid drawn-out mandatory conversations where possible. Try to separate essential information and story progression from flavour and side stuff and make the essential compact and condensed. Then...
Present Non-Essential Story as Offer, Not Obligation
All the nice stuffing in terms of character and story that the player doesn't absolutely have to know (even though they should, because it makes the game better) should be shown as an offer they can accept or ignore.
For example, if an automatic conversation is triggered, because something important is happening, deliver the essential in a few lines, then present dialogue options that let the player ask further questions or end the conversation. Allow the player to initiate conversations with their teammates on their own terms instead of triggering it automatically, possibly subtly alert them if teammates have something to say - a single "Can we hurry up? This place brings up bad memories..." is quickly ignored, but also sets a mood and lets the player know that there may be a conversation to be had. If you have voice lines, Philipp made good points about them on the Writing.SE and this is one more possible use for them.
Put background information into dialogues with NPCs or items descriptions in the world that the player can seek out if they're interested. Side quests are also allowed to be more wordy, because the player can ignore them. Put breaking points into long swaths of text where the player can choose to continue listening or walk away from the conversation. If the few mandatory things all players will see are presented in an engaging way, players who may at all care will seek out more depth in their own. Small but tasty appetizers and a ticket to the buffet.
It's important for these things to be easily accessible as well. Show the player they can talk to teammates in the tutorial, but keep the conversation short. NPCs and books/items are enough of a genre convention that most players will expect and know how to deal with them. Besides flavour about the world and characters, consider making some of the extra information useful in game - not in a way that makes it mandatory in order to be successful, but hinting at secrets is fair game.
It Is Possible to Allow Self-Expression Within the Constraints of a Character
If you look at games like Deus Ex: Human Revolution or The Witcher series, the main character is clearly defined, but there are still dialogue options for them to express different feelings about stuff. Many of these may not even have consequences in the game beyond a different answer from the conversation partner. However, these small choices still allow the player to feel a closer connection to the character. For example, in Human Revolution Adam is asked how he feels about the extensive modifications done to his body without his consent. His answer doesn't really change anything, but this allows the player to consider what they would feel in his shoes and develop empathy.
Of course you can bolster such decisions with in-game consequences. You could, as is often the case in JRPGs, implement a relationship score for your party members that may influence their powers or unlock new conversations and/or something like Mass Effect's companion quests. You can give better rewards for quests if you treat people kindly, find secret solutions or understand people's needs and desires. You can reward playing a character the way you have written them by giving better rewards/relationship points/stronger powers when adhering to their given personality. But you don't have to. Even a small conversation choice with three options and three different answers to them and nothing else can help the player grow closer to the character, as long as the option is well placed and written, i.e. makes the player actually think about the characters.
Intertwine Mechanics and Characters
This is sort of the holy grail of video game design and there are lots of approaches to it.
In terms of a JRPG, you could for example develop the trope of personality reflecting skill sets (brash fire mage, gentle healer etc.) and have characters react to their abilities. A gentle character may feel uncomfortable having destructive abilities and relieved to learn supportive ones. There's a world of difference between a voice line of "I'm sorry I have to do this." and "Burn, baby, burn!"
A typical shounen protagonist would probably think conjuring a big firestorm is awesome as hell while a more subtle power is lame and boring. However, learning how to use such powers may also change their ways of thinking and influence their personality in turn.
Not to mention that it makes perfect sense that to master a certain skill you have to actually understand it, which may require character growth on its own. Think of Avatar: The Last Airbender and the scenes in which Zuko attempts to learn to bend lightning, but his inner turmoil keeps him from achieving the necessary control. Later, Zuko grows as a character but loses his firebending powers, because they're fuelled by rage, and he goes out to learn a new, more stable style of controlling fire, that is based on understanding fire as source of life instead of destruction.
Character development that makes your characters more powerful and your playstyle more fun (always remember: numbers are boring, options are interesting) incentivises players to follow the story, who are otherwise more interested in gameplay. Just try not to present the story as a necessary evil or price to pay to get to the fun stuff.