Speak / chant throughout the ritual
Use Latin, Tolkien Elvish, or another foreign or created language. Learn a set of specific phrases that are relevant to the ritual that you can chant over and over, as well as other phrases that should be said at specific action points during the ritual. Chanting is a standard part of many rituals for good reason. Be consistent with your chanting, and consider rehearsing your chants by themselves in addition to with the rest of your ritual.
Build a rhythm
Your chanting and motion should make the ritual feel as though it has a certain rhythm, or beat. This can change over time - speeding up or slowing as the ritual reaches completion - but it is important that everything you do in the ritual follow that same flow.
Use your hands
If you are not using your hands to perform some other action in the ritual, keep them flowing as you "channel" the magic.
Move around, using your body as a conduit for the magic that you are building. This is particularly relevant if you have a large ritual area. Dance and move in tune with the rhythm of your ritual.
Many props can add a feel of authenticity to a ritual. If your game already requires the use of "components," represent them with appropriate ritual props rather than the bare minimum. Use an hourglass instead of a stopwatch. Use as realistic of a scroll rep as you can, if you have scrolls. Cast near, or around a fire. You can use candles to represent the flow of magic, or you can use glow sticks - either tiny, 1" ones, the much larger wand sized ones, or even the normal 6" sticks. Drums and other percussion instruments can help immersion a great deal.
Wear appropriate costuming
Wear appropriate costuming for your ritual. In some rituals, dark, rune-covered robes would be appropriate. In other rituals, it might make more sense for you to wear a fur mantle and a bone necklace. Base this off of your character's culture as well as off of the type of ritual.
Distinguish your Ritual Area
If you need to use a circle, fill it with runes. If you can't use chalk, make or purchase a cloth circle that already has runes embroidered on it. If you are going to be particularly mobile, make sure that the general area that you will be performing your ritual in is still distinct from other areas.
Base the Ritual on the Effect
Base the script for the ritual upon the effect you are attempting to cause. As an example, if you are casting a ritual to render a shield indestructible, role-play that you are magically reinforcing the shield. If you are enhancing your companion and granting him great strength, role-play that you are infusing his muscles with magical power. If you are destroying the magic within an item, role-play pulling the magic out of it and burning it, or destroying it with a hammer that you formed during the ritual.
Use Common Ritual Elements
In most rituals, it makes sense to role-play the act of harnessing magical power, building it up, or storing it in an item. Figure out what makes the most sense to be common for your rituals, and practice these parts the most. If you perform these parts well, the ritual will already be immersive by the time that you start the custom part of each ritual. That said, this does not mean you should begin every ritual the same way - you can have custom elements at the beginning and end and the common elements in the middle.
If you don't practice, your movement, chants, and other actions may seem unnatural. Practice performing common rituals or any bigger rituals that you might expect to perform.