I have been running on the Mac OS X 10.5.8 and can't seem to do any additional upgrade.
That is the last version of 10.5, so unless you upgrade to 10.6 or greater, that is it for 10.5. MacBooks from 2008 can run the most current version of OS X 10.7, provided your mac meets the minimum technical specifications to run it.
The latest version of OS X is Lion - 10.7.x.
Upgrading will not make your computer run faster. In fact it may make it run slower or not at all because there is something wrong with your current system. Before you even think of upgrading you should think about cleaning up your system by, perhaps, reinstalling Leopard:
How to Perform an Archive and Install
An Archive and Install will NOT erase your hard drive, but you must have sufficient free space for a second OS X installation which could be from 3-9 GBs depending upon the version of OS X and selected installation options. The free space requirement is over and above normal free space requirements which should be at least 6-10 GBs. Read all the linked references carefully before proceeding.
1. Be sure to use Disk Utility first to repair the disk before performing the Archive and Install.
Repairing the Hard Drive and Permissions
Boot from your OS X Installer disc. After the installer loads select your language and click on the Continue button. When the menu bar appears select Disk Utility from the Installer menu (Utilities menu for Tiger, Leopard or Snow Leopard.) After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list. In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive. If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the installer. Now restart normally.
If DU reports errors it cannot fix, then you will need Disk Warrior and/or Tech Tool Pro to repair the drive. If you don't have either of them or if neither of them can fix the drive, then you will need to reformat the drive and reinstall OS X.
2. Do not proceed with an Archive and Install if DU reports errors it cannot fix. In that case use Disk Warrior and/or TechTool Pro to repair the hard drive. If neither can repair the drive, then you will have to erase the drive and reinstall from scratch.
3. Boot from your OS X Installer disc. After the installer loads select your language and click on the Continue button. When you reach the screen to select a destination drive click once on the destination drive then click on the Option button. Select the Archive and Install option. You have an option to preserve users and network preferences. Only select this option if you are sure you have no corrupted files in your user accounts. Otherwise leave this option unchecked. Click on the OK button and continue with the OS X Installation.
4. Upon completion of the Archive and Install you will have a Previous System Folder in the root directory. You should retain the PSF until you are sure you do not need to manually transfer any items from the PSF to your newly installed system.
5. After moving any items you want to keep from the PSF you should delete it. You can back it up if you prefer, but you must delete it from the hard drive.
6. You can now download a Combo Updater directly from Apple's download site to update your new system to the desired version as well as install any security or other updates. You can also do this using Software Update.
Be sure you have at least 12-20 GBs of free space on your hard drive for the above. An even better solution is to make a backup of your current system to an external hard drive. Then go to your local Apple retailer and purchase a copy of the Snow Leopard DVD. Boot from the Snow Leopard DVD then erase your hard drive and install Snow Leopard fresh. Then download the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8. You can now access the Mac App Store to purchase and download Lion which will then bring you up to date. After all this you can begin restoring your files from your backup.
Then there's practically nothing you can do. Here's what might try:
Boot into Safe Mode then reboot normally. Then follow what you can from this:
Kappy's Personal Suggestions for OS X Maintenance
For disk repairs use Disk Utility. For situations DU cannot handle the best third-party utilities are: Disk Warrior; DW only fixes problems with the disk directory, but most disk problems are caused by directory corruption; Disk Warrior 4.x is now Intel Mac compatible. Drive Genius provides additional tools not found in Disk Warrior. Versions 1.5.1 and later are Intel Mac compatible.
OS X performs certain maintenance functions that are scheduled to occur on a daily, weekly, or monthly period. The maintenance scripts run in the early AM only if the computer is turned on 24/7 (no sleep.) If this isn't the case, then an excellent solution is to download and install a shareware utility such as Macaroni, JAW PseudoAnacron, or Anacron that will automate the maintenance activity regardless of whether the computer is turned off or asleep. Dependence upon third-party utilities to run the periodic maintenance scripts was significantly reduced since Tiger. These utilities have limited or no functionality with Snow Leopard or Lion and should not be installed.
OS X automatically defragments files less than 20 MBs in size, so unless you have a disk full of very large files there's little need for defragmenting the hard drive. As for virus protection there are few if any such animals affecting OS X. You can protect the computer easily using the freeware Open Source virus protection software ClamXAV. Personally I would avoid most commercial anti-virus software because of their potential for causing problems. For more about malware see Macintosh Virus Guide.
I would also recommend downloading a utility such as TinkerTool System, OnyX 2.4.3, or Cocktail 5.1.1 that you can use for periodic maintenance such as removing old log files and archives, clearing caches, etc.
For emergency repairs install the freeware utility Applejack. If you cannot start up in OS X, you may be able to start in single-user mode from which you can run Applejack to do a whole set of repair and maintenance routines from the command line. Note that AppleJack 1.5 is required for Leopard. AppleJack 1.6 is compatible with Snow Leopard. There is no confirmation that this version also works with Lion.
When you install any new system software or updates be sure to repair the hard drive and permissions beforehand. I also recommend booting into safe mode before doing system software updates.
Get an external Firewire drive at least equal in size to the internal hard drive and make (and maintain) a bootable clone/backup. You can make a bootable clone using the Restore option of Disk Utility. You can also make and maintain clones with good backup software. My personal recommendations are (order is not significant):
Visit The XLab FAQs and read the FAQs on maintenance, optimization, virus protection, and backup and restore.
Additional suggestions will be found in Mac Maintenance Quick Assist.
Be sure you have an adequate amount of RAM installed for the number of applications you run concurrently. Be sure you leave a minimum of 10% of the hard drive's capacity as free space.
Without your installer disc you cannot repair your hard drive or permissions which is something you need to do.
Do you have a bootable backup?