So, I've been working on a mod for a fair bit, and I finally got around to working on capabilities, which are linked to the player. However, whenever I try to actually register one using 

CapabilityManager.INSTANCE.register(IClass.class, new Storage(), Class.class);

It states that the register function is deprecated.

Heading over to the actual CapabilityManager file I find this:

@Deprecated
public <T> void register(Class<T> type, Capability.IStorage<T> storage, final Class<? extends T> implementation)
{
    Preconditions.checkArgument(implementation != null, "Attempted to register a capability with no default implementation");
    register(type, storage, () ->
    {
        try {
            return implementation.newInstance();
        } catch (InstantiationException | IllegalAccessException e) {
            throw new RuntimeException(e);
        }
    });
}

The file is stating that the register is deprecated, and will be fully removed for 1.13. Is there some alternative to what I am doing? If so, how does one go about doing it?

Use the non-deprecated overload that takes a Callable argument.