Right now, I have a rocksmith cable and it does connect my guitar to my PC perfectly fine, I can play with "Guitar Rig 5" but it does cancel every other sound coming out of the computer. So, I can either listen to something or play the guitar, not at the same time. I think this is a limitation of my on-board sound card(chip).
Are you using Windows and ASIO drivers? Because ASIO has that limitation, unless you use an ASIO multi-client. See here: How to make ASIO work with multiple programs: how can I listen to Guitar Rig plus something else through the same interface?
You might be able to fix the issue without buying a new interface.
I am in need of a very basic audio interface (preferably internal) to plug my guitar into my computer. (...) I know there must be a lot of external and quite expensive solutions to this but I am only looking into the most basic and cheap option (under 100$).
Do not go internal. (unless you have another important reason to do so, other than the price)
1) External interfaces are more flexible.
2) Cheap internal interfaces tend to have severe noise issues.
3) External can be as cheap as internal
You can get an external for as cheap as $30 USD (I wouldn't recommend that one, though). The cheapest internal at sweetwater is at $179 USD.
You can still get a not-terrible-interface with instrument (1/4" TS) input and/or mic/line combo input for under 100 bucks.
For example, the Behringer U-Phoria UM2 gives you combo mic/line input and an instrument input, preamps, and phantom power for $50 USD. I think the downside is that it might be able to record at 16 bits only (which is very common in that price range), but if you don't mind recording/playing at 16 bits instead of 24, that's not an issue.
For 10 bucks more, you can get a Lexicon Alpha: 2 ins (mic and instrument) with preamps and phantom power for $60 USD, and this one can record/play at 24 bits.
(this 24 bit thing is specially useful while recording)
From there you have countless options all the way to the 100 bucks limit. Look for something with an instrument input for your guitar, that can record at the very least at 24 bits and 48kHz, that can handle something that you might want to implement in the near future (mic, maybe a free DAW included, maybe more than 2 outputs, MIDI in/out, etc).
Go to a page like seetwater, and visit their external audio interfaces section, and filter by price from $25 to $100 USD. You'll see several options for under $100.