If you're trying to replace the functionality that existing VBA and iLogic routines did you don't need the iLogic automation class at all. The automation class is really for manipulating rules. All the rest of the utility that iLogic provides is short hand to the API. Just use the Inventor API directly. Some tasks may be more verbose and explicit. It will be very difficult if not impossible to try and use iLogic methods from an add-in. But don't worry. There's nothing mystical to using the API. You just have to approach things in much smaller chunks of functionality. Note: VBA uses the same API so most of those should directly port. You'll just be porting the code from VBA to VB.NET and maybe setting up the context in which it is run.
For example, suppressing an object.
iLogic Rule Method:
Component.IsActive("Part1:1") = False
From an Add-in (Absolutely no error checking, this will normally be broken up.)
Dim comp As ComponentOccurrence = invApplication.ActiveDocument.ComponentDefinition.Occurrences("Part1:1")
comp.BOMStructure = BOMStructureEnum.kReferenceBOMStructure
comp.Suppressed = True
How to find the info:
First you have to understand what the iLogic rule and methods are doing. The Component.IsActive method takes the name of a component, finds it in the active document, changes the BOM structure to Reference then suppresses the component.
Using the Inventor API help file for reference. I know I want to affect components. Looking through the API under the topic "Building an Assembly" I see how an assembly is structured.

I pick the ComponentOccurrence Object cause it's a singular object low on the list that sounds like it could represent a component.
Now I have to see how to get to it. The diagram gives me the hiearchy to get to it. Using the API I look up each object and find the property that exposes the next desired object. I have to always though start from the very top which is the Inventor Application which I've previously defined at the start in my addin.
Thus I get: invApplication.ActiveDocument.ComponentDefinition.Occurrences("Part1:1")
Now that I have a component I look at the ComponentOccurrence object in the API to see what I can do from there. (Alot) But all I'm interested is seeing how to change the BOM structure type and suppressing it. There I find the properties, BOMStructure and Suppressed. The help file tells me what kinds of values they can be set too.
So you use that process you use for basically everything you want to do. And ask lots of questions here. The API help file will become your best friend.
Also best to completely ignore iLogic and iLogic.automation while developing add-ins unless your add-in automates iLogic Rules and only the rules themselves, not what the rules are doing.