You need to use the EPiServer Deployment Center. And if I remember correctly, you need to go to 5.2.375.7 (R2) before you go to 5.2.375.236 (R2 SP2)

There are some breaking changes from R1 to R2. E.g. the access level is not checked on GetPage or GetChildren. See all the breaking changes at http://world.episerver.com/en/Documentation/Items/Release-Notes/EPiServer-CMS/EPiServer-CMS-5-R2---Release-Notes/Breaking-Changes/

Make a backup of your project files and your database and do a local test-run to see how it goes. It's hard to say what may go wrong without knowing your code :-)  A "simple" solution without many changes will upgrade without much trouble.  

Can you just verify what you said here as you suggest going to v5.2.375.7 before 5.2.375.236, is this not the other way around or is there a typo in the version numbers?

Is there a support page that helps with this process? I need to know where to get the individual version files from and how to go about upgrading the EPiServer code even before I start fixing any customised code changes that we have made.

If you install the EPiServer versions R2 (5.2.375.7) and R2 SP2 (5.2.375.236) and open the deployment manager, you will only have the upgrade from R1 SP2 (and SP3) option available in the R2 version. The R2 SP2 upgrade option is only available for previously installed R2 sites. (or sites upgraded from R1 to R2)

I tried usingEPiServer Manager 2.0.19.10-2 to upgrade the website on localhost but get the following error:

Import of sitedata failed: An error occurred creating the configuration section handler for episerver: Could not load file or assembly 'EPiServer.Configuration, Version=5.1.422.267, Culture-neutral, PublicKeyToken=8fe83dea738b45b7' or one of its dependencies. The system cannot find the file specified. (C:\ my file path \web.config line 5) 

Had a bit more luck using EPiServer Installation Manager in v.5.2.375.7 and upgrading site from 5.1 SP2

It starts to work, but then i get the following error

Error - System.Management.Automation.CmdletInvocationException: Access to the path 'all-wcprops' is denied. ---> System.UnauthorizedAccessException: Access to the path 'all-wcprops' is denied.
at EPiServer.Install.InstallationManager.DoInstall()
at EPiServer.Install.InstallationManager.CommitBulkInstall()
at EPiServer.Install.Common.Cmdlets.CommitBulkInstallCmdlet.ProcessRecord()
at System.Management.Automation.Cmdlet.DoProcessRecord()
at System.Management.Automation.CommandProcessor.ProcessRecord()
--- End of inner exception stack trace ---
at System.Management.Automation.Internal.PipelineProcessor.SynchronousExecuteEnumerate(Object input, Hashtable errorResults, Boolean enumerate)
at System.Management.Automation.Parser.PipelineNode.Execute(Array input, Pipe outputPipe, ArrayList& resultList)
at System.Management.Automation.Parser.StatementListNode.Execute(Array input, Pipe outputPipe, ArrayList& resultList)Copied ElektroPost.Licensing.dll 

The migration is a bit tricky. The tool doesn't recognise version 5.2.375.236 and yes most people working in software houses would class .7 as being more advanced than .236 at first glance.

Finding this file version   5.2.375 .7   so that we can upgrade a 4.62 site is a nightmare.  Having to go to SP1 and then upgrade from that... It would be simpler if the verification looked at the 5.2.xxx.xxx bit only when verifying.

Anyway - make sure you have whatever files it wants as DLLs somewhere it can reach them.