Even though Cosplaying is primarily dressing up, acting as your character for LARPing or just for fun is a good way to make your cosplay memorable to others. To effectively pull off your character, it helps to use these methods as a way to familiarize yourself with your person/creature.

Do your research!!
Before you can play a character, you must get to know them first! Watch or play anything they are involved in, and study them carefully. Get to know the basics about them: personality, movement/walking style, accent, background, etc. Wiki pages help  a lot with this sort of thing.

Use a character sheet!
Whenever I potray a character for a play/production, I have found that these Character Development Sheets help me get to know my character way better than simply reading a script or playing a video game. The questions are specific enough that you can't depend on the character's origin for answers, so it requires you to "become the character" and put in answers that you believe fit best.

Learn to act off script!
Extremely important with cosplays/LARPing, acting off script is when you don't depend on a script or show/video game in order to potray your character. At a convention, people will not be satisfied with your character if you just act out the same scene over and over. You must learn to be your character without any guidance or preset responses. In other words, think of an interaction between you and another person as an Improv scene.

Play your character, but play them naturally.
The most important - and most difficult - aspect of acting is to not make it look like an act. You should come off as the character you are playing, not as someone attempting to play your character. For example, most ordinary people  will imitate a famous person and it will look so overdone that you know they are faking it. However, a good actor will imitate a famous person so well that you may temporarily forget that they are not the person they are potraying.

Before the convention, be your character for the day!
A good way to practice your cosplay/LARP is to be that person for a whole day. This is best suited for a day that you are stuck in the house, as you may not be confident enough to act like your character in public. From the time you wake up to the time you go to bed, move and act like your character would on a regular day. Dressing up is optional.

Leave some extra advice you may have in the comments!