This assumes, you have your disk on /dev/sdc1 your configuration may change.
Also, if you want to convert your root partition, you will have to boot with another Linux, you will not be able to convert your running root partition.
sudo tune2fs -O extents,uninit_bg,dir_index /dev/sdc1
Remember to change /dev/scd1 to fit your needs.
sudo fsck -pf /dev/sdc1
Once again remember to change /dev/sdc1 to fit your needs. Now you can mount your new system as ext4
sudo mount -t ext4 /dev/sdc1 /mnt
And you will have to change in /etc/fstab file:
Where it said ext3 now should say ext4 for the /dev/sdc1 partition.
Before:
/dev/sda1 / ext3 defaults 1 1
After
/dev/sdc1 / ext4 defaults 1 1
Your options may differ.