To answer your question, really, I feel it's prudent to walk through
the deployment steps and call out where places that might be gotchas.
I'll finish with a summary of what I would do.
Sitecore 7.5 - 8.0 Initial (141212)
The first step that you'll want to do is upgrade to the 8.0 Initial release. You can find the Upgrade Guide here. During this upgrade, there is not much that is changed on the xDB front, however Step 1.2.8 - Redeploying Marketing Data is going to be taking portions of your analytics database and redeploying it to a new schema.
With regard to xDB there is not much going on here. Also worth noting that schema changes in xDB kind of happen automatically, one of the pro's of using a schema-less data repository.
Sitecore 8.0 (141212) - 8.1 (151003)
The second step is going to be upgrading from 8.0 to the inital release of 8.1. You can find the Upgrade Guide here. This is a pretty involved upgrade step, mostly doing a lot of work on the SQL side, however, really starting to augment the xDB side of the house with content testing data in xDB.
You'll want to pay careful attention to Step 1.3.3 - Upgrading Content Testing Data which has direct impacts on xDB and xDB configuration.
Sitecore 8.1 Initial (151003) - 8.1-Update 3 (160519)
The third upgrade step is going from 8.1 Initial to 8.1 Update 3, again you can find the Upgrade Guide here. In this particular upgrade, there are no specific xDB upgrade steps, short of making sure that you disable xDB in the config while the upgrade package is running.
How I would do it
1) I would look into making a copy of your Mongo collections. To do this, as long as you are using MongoDB 2.1+, you can copy a collection using the following syntax in a tool, like Robomongo:
In this example, the collection I want to copy is example1 and this will copy it to a new collection called example2. You'll want to do this for all of the mongo collections:
2) I would then setup your second 7.5 site (upgrade environment) that you'll be using as your upgrade site and point the connection strings for mongo to the new collections.
3) Perform all of your upgrade steps in your upgrade environment.
4) Adjust/refactor your custom code base as needed.
Now, in a perfect world, you would think that switching over IIS to the new site would work, and maybe in your case, the rate of change in your production content is minimal enough that this work. If that's the case, the I think that's all you need to do.
However... if the rate of change is high, you may be looking at having to upgrade the Content Management environment in PRODUCTION again. Only this time, you know what to expect, you've done it once before in the upgrade environment, and now it's just a matter of doing the work. This will also include any changes to production xDB.
At that point, use your standard deployment methodology to move code bits to the CD servers and any other role servers you might have deployed. And your off to the races.
You really don't have to worry too much about xDB. I think the Content Testing changes are the only point that you'll want to pay attention to. The rest of the upgrade, for the most part, has xDB disabled.