Airport Extreme and Express integration.

In addition to the information provided by Tesserax, it is important to remember that Apple will only allow you to "extend" a network wirelessly one time. By that, we mean that while it is possible to have more than one extender working on a network....each extender must connect directly to the main base station in order for the network to be extended.

So if you look at the network layout in AirPort Utility, you can learn a lot about how well your network might actually be performing and extending the network.

If things are working correctly......the four Express devices will all be lined up horizontally side by side with a clear indication that that each one of them has a direct connection to the main router......in this case, your AirPort Extreme.

Or stated another way, you would not want to see any Express that is displayed under another Express connecting to another AirPort Express. If you do see this, the "bottom" Express is not extending the network at all, despite what you might think. And, any other WiFi devices will of course not be able to connect to that Express, since it is not extending the network.

Apple explains this in a slightly different way...but the result is the same. Unfortunately, their example does not display things the way that AirPort Utility displays things. in Apple's example below the AirPorts would all be displayed vertically.

Important note

If another extended Wi-Fi base station ➋ is placed between the primary Wi-Fi base station ➊ and the extended Wi-Fi base station ➌, the extended Wi-Fi base station ➌ will not allow clients to join it. All extended Wi-Fi base stations must be in direct range of the primary Wi-Fi base station

Can I plug my device via cat5 into my already configured booster to ensure that my PS4 is connected to the booster?

Yes.

...how can I move my device from using the extreme to using the booster that has a much stronger signal?

The overall issue is using a wireless connection between the AirPort base stations for extending your Wi-Fi network is prone to overall bandwidth performance issues. That's because of two reasons:

1. Extending the network in this fashion, reduces the bandwidth by half at each of the extending base stations.
2. The extending base station can only extend the network at the same bandwidth at the same level as it receives it. This plays heavily on precise base station placement. Please see the following user tip for more details about this: AirPort - Optimal Base Station Placement - Apple Community

The bottom line? To get the maximum bandwidth for streaming and gaming, you need to use Ethernet connections between your base stations. In addition, the AirPort Express base stations only sport 100 Mbps Ethernet ports. You would need to upgrade them to either the Extreme or Time Capsule models to gain 1000 Mbps ports. Unfortunately, as you may already know, Apple is gotten out of the networking hardware business and finding new units will be more difficult, if not impossible, over time.