You will need to firstly download a copy of your contacts from your Gmail account to computer using a web browser on your PC or Mac. Then upload the contacts from computer to Hotmail account also using a desktop browser. Check out details as follows.
You can refer to the following guide for the instructions, see how to export contacts from Gmail? It is very easy to follow.
Go to log on your Hotmail account from this link http://hotmailsigninaz.net/mail/hotmail/create-signatures/ , it opens up your contacts page, the Outlook People tab. And you will see a screen like this:
Go to the Manage menu from the above screen and choose Import contacts, you will see a list of all supported email types including Gmail, Outlook, Yahoo Mail, Windows Live Mail. Choose Gmail to proceed to the next screen as shown below.
Click Browse button from above screen to locate your saved Gmail contacts backup on your computer and import the contacts from Gmail backup to your Hotmail account.
If you want to transfer your contacts from the other way around, see this guide: how to transfer contacts from Hotmail Gmail?
Update April 2018: this article was originally published in 2014 and has since been updated. in this update, the Gmail to Outlook contacts import process was updated to reflect the change of Outlook for web.