Some fun things to think about:
Most ESL speakers learn the language from media. There's that old trope of learning how to speak by watching cartoons and reading comics, giving the character a pretty wacky vocabulary and syntax. You can go with that.
You could have the aliens studying, and believing they've mastered, English. But we all know English makes zero sense. It's full of contradictory rules. Depending on the style of your aliens, a haughty "I follow logic and understand all rules" type ET would THINK they understand it, and would refer to deer plural as "deers" and argue about it because that's what makes sense given the "rules" of English.
The universal translator trope is a trope, but a useful one. Depending on where the focus of your story is, you may WANT to focus on the language barrier. If you just need to get past it to get to the actual plot, you can use the nanites/microbes/braincomputer/telepathic power trope that lets people understand each other so you can get on to the meat. I think deciding if you want to make this a part of the human/alien interactions, or whether you can do without it, will help inform your decision.