I have included my character's home brew character sheet as an example of how I am trying to keep track of the stuff I talk about below. I realize that it has way more detail than you really need, but it took very little time to think about my character and scribble out these notes. Plus, it was fun.
Know your abilities - This is a mechanical one, but a critical foundation for everything else. Know your character's class abilities, racial abilities, skills and other things that makes your character unique. This allows you to play the game well. Not knowing the rules that specifically affect your character would be like playing checkers without knowing that getting your piece to the opposite baseline allows you to say "king me."
|Merka, page 1, the critical stuff|
The home brewed 5e character sheet shown throughout this post is the format that I have developed over the course of the Shearingvale campaign. It was specifically created to allow for the important mechanical things to be highlighted and readily available on page 1. The different classes have morphed into their own semi-custom sheets, and this is the one for a paladin - mainly fighting but with some spell casting. Merka's important stuff is highlighted. Racial abilities for Savage Attacks and Relentless Endurance are noted, as well as paladin abilities like Lay on Hands and Divine Sense. My fighting style of Great Weapon fighting is noted. I also don't want to forget that I am Stealth-disadvantaged because I am wearing chainmail armor. Additional details on some of these abilities are shown on page two of her sheet.
|Merka, page two, secondary info|
Know your character - Less important than the mechanical knowledge of your character, but no less valuable, is a sense of what your character is all about. What do they believe in? What do they like? Dislike? What scares them? Do they have a behavioral quirk or two?
|Merka, page 3, the character fluff|
I'm not saying that you need to write extensive back story and all of that (although if you want to do that for your own fun, then by all means do so, as long as you do it with the full knowledge that you are only doing this for yourself and it is very likely that nobody else cares...). What I mean is to have enough of an idea of what your character believes as a person to allow you to react accordingly (and interestingly) during in-game situations.
Exploit your strengths - Understand what you are good at and find ways to leverage those abilities (no matter how minor). This is the extension of knowing your abilities and the rules for them, and really is just the implementation of that knowledge. Look for every opportunity to leverage the skills that your character has that makes them special, and therefore valuable to the team. If you are a cleric and your group is hard-pressed by attacking skeletons, you shouldn't forget that you can attempt to Turn Undead. This is directly related to the next point...
Enable your friends' success - This is the flip side of exploiting your character's strengths, and is about being a good player on a personal level in terms of interacting with your friends at the game table. Try to remain aware of what your friends' characters are good at, and be mindful of allowing them to shine in those circumstances. One of the traits of a good DM is to spread out the opportunities for the different characters to take the spotlight and have their moment. Players should be mindful of that for each other as well.
|Merka, page 4, notes and reminders|
Embrace your weaknesses - This is all about role playing, and has nothing to do with the mechanical aspects of the game. Identify a few things that you aren't good at (even if there aren't rules for that thing), and a few things that scare you. Let yourself be afraid of heights, or too scared to go in a boat, or be terrified of horses, or overly fond of a high stakes card game. This will make your character seem like more of a real person, and more importantly, will give an attentive DM the hooks they need to put you in uncomfortable situations. This is often what creates the memorable moments. And it gives the DM material to work with, which they will appreciate.
In my Shearingvale campaign, we had a situation in the Underdark where our dwarven cleric announced himself afraid of heights when the party arrived at a narrow railing-less bridge over a deep chasm. This had no affect on the game, but made the scene much more real than "everybody roll a DEX check and just don't roll super-low".
Listen - Practice your active listening skills. Note the tidbits that the DM is throwing at you. Scribble a few quick notes (they don't need to be extensive). Pay attention to what the other players are saying so that you know what matters to them. Good role playing is mainly about riffing off of what the DM and other characters are saying and doing. You can't do that if you aren't paying attention. A well played and memorable game has a lot more depth to it than "is it my turn to attack yet?"
I'm sure other thoughts will occur to me, and I suspect I will add to this short list as thoughts percolate.