I've seen similar questions on here but often people use short hand when they are chatting like "U" and "R" and how would you go about this using screen names? Or a public chat with multiple strangers and adding in random people for the effect? I seem to be having trouble doing this.
The first thing that I would do is defining a bunch of writing styles which are used by different people in the chat. Some might care for correctness, while others do less so or not at all, and even then they can write things differently; For example, the phrase "I am" can be rendered as: I am, i am, I'm, i'm, Im, im; Some might only use/prefer one specific variation while others use several on different occasions. Other things you can variate:
number of typos
use of smileys
AE/BE, and dialects in general
use of uppercase letters
general length of messeges (like some tend to write longer messeges)
period of time someone needs to answer (some people type slowly)
Also be sure to have at least one person who can't be bothered to use backspace at all.
It really depends on your novel, and the chat platform your characters use. Every chat platform is different. Some characters might use their real name, others will use a screen name. Whatever they choose for their screen name will ALWAYS say something about who they are as a person.
Whether characters use text speak, abbreviations, line breaks between each sentence, and emojis will tell the reader a lot. Characters may also use line breaks to emphasise a point.
If it works for your character to write U and R, then go for it. It can be jarring to read, but if it fits your character, do it. Just be cautious of long passages of text using it.
Look at other books that use similar concepts. The only ones I can think of right now are some of Meg Cabot's - The Princess Diaries and Every Boy's Got One. The latter is written purely in diary entries, emails, and BBMs. While BlackBerries are hugely out of date now, it's always interesting to see what other authors do.
There's also the option of using triangular brackets to break things up as some chats do. For example, < Liam entered the chat >. It can also be used to type an emotion: < sobs >. This is often used in gaming communities.
Having had a similar predicament deciding how to format text messages in my novel, the best advice I can give is to experiment. Try as many variations as you can to decide what works.