As @Ilan mentioned, if you need to manipulate pixels, Photoshop's the way to go. Illustrator is for vector graphics.
There are multiple ways of making a selection, but in your case the Quick Selection Tool (W) does a decent job:
- click on the bear itself and drag outwards little by little and make sure the selection is only within the bear shape (even if it doesn't include every since hair edge)
- in the top bar select Refine Edge and play with the parameters until you get the results you're after (exclude the background without loosing too much of the hair detail)
- hit Ctrl+J/CMD+J to copy the selection to a new layer
Here's a what I got in a fairly short time:
Notices that the right side isn't perfect. The photo is slightly over exposed and it's hard to separate the hair highlights on from the bright background, but a bit either pre-processing tones or post selection cleanup can help if necessary.