It's bad enough that lint and such can get 'hung up' in a dryer vent duct. Add to that a bit of grease and residue from a microwave over time, and you'll have either a nasty recipe for a fire or a blockage of the duct.
You'll either need a separate duct for you project or somehow tap into the existing kitchen duct.
My best to ya and hope this heps.
Jay J -Moderator
PS: If you're in a Rental, you may need the 'approval' of the Landlord. Do realize that if you do this project on your own, and in the process (complete or not), there's holes in walls or damage, you may be 'charged' to have it fixed/put-back-to-normal ...