I see my antminer has cgminer built into it. Do I need to have cgminer on my computer screen while "solo mining" for bitcoins ? or can I just have the antminer on and hashing straight through the router ?
cgminer is just the client software to control your ASICs. Where cgminer gets it's work from is your question and that depends ...
- You can run your own full node and point your hashing power there
- or way easier would be to just use some hosted solo pool like http://solo.ckpool.org/
On the website you finde further information on how to solo mine.
The basic idea of mining is that there is a node which has work (actually calculating hashes) to distribute to (usually a pool of) workers. Your Antminder would typically be such a worker, i.e. it is usually configured to ask a pool for work, do work and report results back.
The idea of solo mining typicalls means to run your own local pool on your machine (which of course doesn't charge you anything) and run just one worker.
What that means related to your question:
Theoretically, an Antminer ist a computer after all, so it should be able to run a node which distributed work to itself, which is what solo mining is all about. Just I haven't seen any option to enable this in the Antminer GUI. This maybe due to the fact that if was probably academic anyway as solo mining will not generate any coins for you anymore. Also it may be an issue if your Antminder would have sufficient storage capacity to run a node.