install new SSd drive in HP desktop pc

I have tried to install a new SSD drive in my HP CQ5226uk desktop pc. I intended to install alongside the existing 350GB sata drive. There does not appear to be enough space for a 2nd drive of any sort inside the metal caddy containing the sata drive. The only available space appears to be in the area above the DVD drive and that is an area suitable to install a 2nd dvd drive.
You can install an SSD anywhere it will fit. Some cases have a notch in the back of the motherboard tray! But because it is not a mechanical drive there is no concern about vibration, so you can tape it down, or use zip ties, if you wanted. There is no need for it to have it's own bay in the hard drive caddy.

Also there does not appear to be any additional connections for a 2nd HDD drive of any kind. I can see only the black connectors attached to the existing drive.
This is a problem. If you do not have an extra SATA connector and SATA power connector to use you are out of luck as long as you own that motherboard.
Note: you can buy and add a SATA controller PCI-e add on card to gain some extra SATA ports. But in that case be sure to attach the SSD directly to the motherboard and the HDD to the card.

Assuming I can find a way to connect the SSD drive, Do I have to change anything in the BIOS, if so may I have the details please.
Normally, no. In SATA a hard drive is a hard drive. It will be recognized by the BIOS without intervention.

Drivers for SSD drives I have a disk but im not exactly sure when & how to install the drivers or can windows find them when the ssd is connected.
Windows will install it's own driver. The disk will contain hard drive utilities for managing the SSD. These are optional. It may also contain an Intel Rapid Storage Technology Driver that could be beneficial if you do not already have one installed. Use the Custom install option (if there is one) and install only what you need or want. Ask here if you don't know what anything is.

Finally if i am unable to install this drive would it be possible to use it as an external drive with the OS loaded on to it. (Although its only 120GB it cost a lot of money)
Not advisable. Running the OS from an external drive is an iffy proposition.

In addition to the previous explanations you may want to have a look at this:

SSD - Install and Transfer the Operating System

There is one point above where I disagree - running an OS from an external SSD is working very well, especially if you have a USB3 attachment. I run a whole collection of operating systems (Windows 8 and Linux) from an external SSD and they work extremely well. Check this:

Portable OS - Carry your OS on an External Drive

And here is a demo running Windows 8 from an external SSD;

VMware Player - a Windows 8 Demo

I just made a little recording of Windows 8 under VMware Player running from a USB2 attached SSD. It will give you a feel how well that runs - pretty well I think. A USB3 port is, of course, quite a bit faster.

Good to know whs! I do not partake in such things so it is good to know it can work. I have just seen some threads where peoples have got problems making it work.

One curiosity though: you say "running an OS" from an external drive. And running them in virtual machines.
The OP would need to run "the OS" from the external. Have you done this also?

But you are right, without USB 3 speed it would be pretty bottle-necked.

Yes, you run the OS from the external drive but as virtual partition of VMware Player - which after all is only a folder sitting on the external drive.

The advantage is total isolation from the particular PC in terms of drivers, activation, etc. VMware Player is the host system. That's why you can carry it around to any system that has the VMware Player base system installed. That gives you the umbrella.