I don't know if this counts as an interpretation exercise, (if someone there is trained in theater or understand dramaturgy, let me know) but when I create and play with a character, I already have a background and a general idea of who and what he should be, so I relax, close my eyes and review the whole story I created for him in my mind, trying to put myself in his place (like the flashbacks in Naruto), I don't know if this is a technique validates that the actors use it or if it’s because I’m very empathic, but it helps me to “get in character” before each game, it also helps me choose which character I’m going to use at that moment, according to my mood! (if I feel like venting a little bit of aggressiveness and violence, I use an Ork, and when I feel particularly sadistic, I choose a corrupt champion or a demon)
The last time I tried to insert myself in the game, was when I created a "gray warrior" (a character that does not specialize in anything specific, usually being mediocre in everything, but that with A LOT of effort, can reach the maximum level in everything too) in CoC1! And even though I managed to complete the game completely, with all the equipment, missions, abilities and related to all the NPCs, it was not as rewarding as I thought it would be!
Then I decided to start over, but this time, I would focus only on a specific NPC, molding my Champion to match it so that one complements the other, and it was a totally different experience! I felt that not only was it a totally satisfying gamble, but also that if I were to recapitulate his entire trajectory, it would be a story that would be worth telling! Since then, I have always created a character for every possible construction in the games!(I chose Amily to start, since she was one of my favorite companions and one of the first ones you find in the game, the result was that I created a mouse swordsman totally focused on speed and critical damage, wearing a white silk shirt with tight leather pants and a Jeweled Rapier!)
So to sum it all up: instead of inserting yourself into the game, let the game insert itself into you!