All you need to do is to create a Visualforce page and drop it in wherever you'd like. Here's an example:
<apex:page standardController="Lead">
<script src="//platform.linkedin.com/in.js" type="text/javascript"></script>
<script type="IN/MemberProfile" data-id="https://www.linkedin.com/in/{!Lead.LinkedInId__c}" data-format="inline" data-related="false"></script>
</apex:page>
Do this, then edit the Lead page layout and drop it in somewhere, and you'll get to see the LinkedIn embedded API.
Or, if you want to actually show the profile, you could do that, too, by a similar technique:
<apex:page standardController="Lead">
<iframe src="https://www.linkedin.com/in/{!Lead.LinkedInId__c}" />
</apex:page>
You'll want to adjust the code to get the experience that you want.
Note: Some sources state that you can't iframe a LinkedIn member page, so this second technique may not work. I've used the first method successfully in the past, though, so that should be fine.
You can read a bit more about adding Visualforce to a standard page layout in this trailhead unit or in the documentation.
As a related note, there is a Profile API that you can use as another alternative, allowing you to query the data from the profile and you could render the results yourself. Again, this can be done in Visualforce.