I have recently purchased the 5e players book and I really like the character system in 5e but my usual DM likes to play 3e and I was wondering, are these compatible? Can I use a 5e character in their 3e game?
The rules are, despite having similar names for things and some similarities in implementation, completely separate. They do not work together, and a campaign has to choose which edition to use.
Not in any meaningful way, no.
Both 3.5e and 5e are largely defined by their character systems. If your DM likes 3.5e, then the exact thing you like in 5e is precisely in conflict with what someone wants from a 3.5e game.
In theory you could convert a 5e character to 3.5e, but that would defeat the purpose of making the character in 5e in the first place.
I think you can, though you might to rule that to take a bonus action the character has to use a full round action. Also compare spells, and if the 3.5 counterpart of a 5e spell has an XP or material cost, then apply it to the 5e spell. You'd have to also add in 3.5-style saves for the 5e classes, based on their 3.5 counterparts, instead of using the ability saves. Then you should be fine.
5e is meant to be weaker and more balance than 3.5, and they do counteract each other. Though as always, it is best to check with the DM first.