Okay, so there are a bunch of these guys and we don't know a ton about them besides that they all love Célimène and they aren't the sorts of people that we want to be friends with. There isn't a lot to work with, but let's see what we have.
This guy is the first one we meet. At first he wants to be friends with Alceste, but he quickly gets shot down. He is our first example of all the horrible people that Alceste is talking about, and we get that idea quickly with his first few words. He says, "And that it's always been my dearest desire/ To be the friend of one I so admire./ I hope to see my love of merit requited,/ And you and me in friendship's bond united./ I'm sure you won't refuse—if I may be frank—/ A friend of my devotedness—and rank" (1.2.6).
Yeah, you know what we love? When someone comes up to us and asks to be our friend. Also, when they name drop how important and fancy and rich they are. That really makes us want to hang out with them.
When it's clear that the job of being Alceste's BFF is already filled by Philinte, Oronte starts reading a poem. It's bad, and Alceste tells him so. This is enough to start a mortal feud/ lawsuit, evidently because rich aristocrats didn't have a ton to do in the seventeenth-century besides make enemies out of really stupid situations. What happened to being BFFs?
Oronte, like the other suitors, leaves when Célimène's infidelity is revealed.
Acaste is, shall we say, not the most modest of men. These are the first lines we hear from him: "I'm young and rich; I can in modesty/ Lay claim to an exalted pedigree;/ And owing to my name and my condition/ I shall not want for honors and position" (3.1.7).
Gee, how could Célimène turn down such a charmer?
Most of what we know about Clitandre comes from Alceste, who makes fun of him to Célimène when he accuses her of cheating on him. We get the impression that Alceste does not think that he is the manliest of men, if you get what we mean. He asks, "Are you in love with his embroidered hose?/ Do you adore his ribbons and his bows?" (2.1.38).
Oh, and Clitandre helps Acaste bring the letter that spells Célimène's downfall.
Basque and DuBois
These guys aren't suitors. They're just comic relief in a play that is already full of comedy. Basque is Célimène's servant, and DuBois is Alceste's valet. They're both a little dim and have nearly no lines. Here's an example of Basque's blundering, "It's a long tailcoat that this fellow wears,/ With gold all over" (2.6.4). He can't be bothered to find out the guy's name, but, hey, he sure looks important.
DuBois is even worse. He tells Alceste, "Not finding you, he asked me to convey/ (Knowing I could be trusted with the same)/The following message.... Now, what was his name?" (4.4.33). He can't remember the name of the guy, or his message, and later we will see that he even forgot his letter.
Ha, ha! Servants are such knuckleheads!