An update is being deployed on Outlook on the Web (formerly Outlook Web Access) to provide a better integration with Skype for Business.
This update will allow you to start IM conversation from the web interface and not only just managing your IM status
The first thing is you need to check if you have the Skype for Business icon at the top right
Then you will be able to see your contact list
To start a conversation, just hit one of the contact
To add contact to the conversion, just hit Add people button shown just below the contact name
You can switch from a conversation to another using the side bar – which will be open completely when you are on top of the + (plus) sign
To close a conversation, just right click on choose Close conversation, you are then getting prompted to confirm your request
The good point is the conversation is being saved in the Conversation History folder as it is the case with the client and Outlook (hopefully this will be the case soon with the mobile client too)