Love collecting bi characters & real-life members of the "club"! (Though admittedly examples are a bit thin on the ground, what with our being non-existent & all.)

FICTION:

• Azhrarn, "Night's Master", "Death's Master" by Tanith Lee [novels]: For all of my darkly-glamorous, decadent, outrageously-hawt, morally-ambiguous-trickster-figure needs. Also a representative of the all-too-rare bi-male contingent!
• Det. Timothy Bayliss, "Homicide: Life on the Street" [TV series]: First bi character on American TV! Plus a cutie, well-acted/well-written, & on a good show (at least till it went in the crapper near the end!).
• Katina "Katchoo" Choovanski, "Strangers In Paradise" by Terry Moore [comics]: SO cool, hot, & badass! While also being endearingly-vulnerable. Plus distinctive for being modestly-endowed (a rarity in comics, where big boobs are The Law) & scorning all femme-y trappings! (long hair aside)
• John Constantine [comics; TV]: Complex (at least during his original incarnation, the "Hellblazer" series-- where he aged in real time!), surprisingly-realistic (despite being a spell-casting warlock!), explicitly working-class, snarky Brit-- what's not to like? Well, other than the fact that his bi-ness was barely acknowledged over "Hellblazer"'s run, but DC even fixed that with the reboot!
• Phedre no Delauney, "Kushiel's Dart" by Jacqueline Carey [novel]: Morally & psychologically-complex, enjoyably-glamorous character with an original and intriguing concept.
• Lady Nyctasia Edonaris, "Silverglass" Trilogy by J.F. Rivkin [novels]: Fun & charming aristocrat/scholar (often living undercover as a cheeky commoner) in a sword-and-sorcery fantasy setting.
• Christopher Ferris, "As Meat Loves Salt" by Maria McCann [novel]: Very believable & appealing, while simultaneously illustrating some of the period (Civil War-era England)'s more interesting social-history aspects.
• Griffin Fiske, "Original Sinners" series by Tiffany Reisz [books]: Delightfully-fun & funny, sexy charmer.
• Dr. Remy "Thirteen" Hadley, "House M.D." [TV series]: Intriguing, layered, cool, & hot.
• Captain Jack Harkness, "Doctor Who"; "Torchwood" [TV series]: Another cheeky charmer.
• Prince Oberyn Martell, "Game of Thrones" [TV series]: Wonderfully-charismatic, larger-than-life, swashbuckling, unabashedly-bi dude!
• Lieutenant Billy Prior, "Regeneration Trilogy" by Pat Barker [novels]: Realistic, nuanced, sympathetic portrayal.
• Lisbeth Salander, "Millennium Trilogy" by Stieg Larsson [novels; movies]: Kickass-cool wish-fulfillment. Wounded, fierce, formidable genius-hacker-loner who outsmarts & generally lays waste to monstrous men. Can she be my girlfriend, please?
• Kalinda Sharma, "The Good Wife" [TV series]: Triple-threat rarity as a roguish (always has something up her sleeve) South Asian (and a non-stereotypical one at that!) "older woman" who's allowed to be wickedly-sexy!
• Darryl Whitefeather, "Crazy Ex-Girlfriend" [TV series]: Practically unique as a bi-guy (all too scarce right there!) who's both presented in non-sexed-up terms and is middle-aged!

REAL LIFE:

• Tallulah Bankhead
• Lord Byron [George Gordon]
• Alan Cumming
• Marlene Dietrich
• Raul Esparza
• Angelina Jolie
• Frida Kahlo
• Florence King
• Tamara de Lempicka
• Edna St. Vincent Millay
• Sarah Paulson
• Aubrey Plaza
• Kristin Stewart

Now for the BONUS QUESTIONS:

Do you feel the characters you see in media accurately represent bisexuality?

Well, to the extent that they're even there to be seen in the first place: HELL NO! When not just flat-out offensive (still dismayingly-common), the best you can usually hope for is a depiction that, while fun and appealing, bears ZERO resemblance to reality: the awesomely-cool, personification-of-sexiness caricature. Bisexuals presented in anything like an authentic manner are virtually unknown. We're always either creeps/villains or exotic fantasies; we're never just people. Much less drawn from a full RANGE of people, including "next-door" types.

Does the media do a good job with addressing and handling bi issues?

They'd have to do ANY sort of job first! And, yanno, acknowledge that we exist. Outside fiction! As REAL PEOPLE! With REAL ISSUES! Some specific to our sexual orientation! And, in particular, the way that it's (mis)treated by society at large.

What would you like to see more in regards to how they create bisexual characters and handle their issues

• Take a break from the default tropes of fantasy-figure on the one hand, or perverted creep on the other. Think outside the box already! Give us OTHER types of bisexuals, please.
• Expanding on the previous point: more MALE bi characters! Who aren't creeps/villains/exaggerated fantasy-figures! Also more of the following: "normal/average" people; middle-aged-and-older; monogamous; in familiar, well-established roles (cop; lawyer; detective; action hero; rom-com protagonist; final girl; etc.). And can we give the "Hot Bi-Babe" (who is, needless to say, always up for a MFF threesome) trope a well-deserved rest now?
• Create characters who are people, first and foremost. Don't have their bisexuality drown out everything else about them.
• However, do NOT be afraid to make their bisexuality an important part of who they are. It's such a neglected topic! Ripe for discovery! Feel free to explore it in depth, with thoughtfulness, creativity, and nuance.
• Can we skip any more of the "plot twists" that amount to, "OMG hetero-active person was also boinkin' somebody of THE SAME SEX!!!" [cue sforzando]? Where bisexuality is played for shock value? It's really exploitative and insulting.
• Other stereotypical crap I could do without seeing EVER AGAIN: us as cheaters; murder victims; psychopathic killers; criminals; "confused"; sex maniacs; weak/pathetic; "exotic"; decadent; rich degenerates; predators; somehow "above" lowly considerations like what genitals people have; sleeping with both sexes cuz we're Master Manipulators^TM; just so self-confident/arrogant that we expect everyone to think we're Hot Stuff; swingers; vampires. (Well, OK, if you use some of these in a way that's justified, and with both sensitivity and skill, I'll allow it.)