The idea is to have two characters, a large-sized race and a small-sized, linked together where the small character is acting like a leech. We wanted to have the small character be extremely handicapped and require the large character for transportation. The small character would 'borrow' blood from the large character for HP and could take health from others to replenish the health pools of both the small and large character or just one of them. Since they are linked through blood though, when one takes damage they both could, and if one is poisoned or diseased, then both would suffer from the ailment.
The small character likely would be a "blood mage".
Each character would be played by an individual player.
How can we make this happen without being generally overpowered?
Best Answer
Since you are asking specifically about D&D 5e, I will not add any comment past this about other systems, as that would detract from a solid answer. Instead I will focus on the system requested.
Update - Guidelines: Wizards of the Coast released a new 'Unearthed Arcana' entry on April 6, 2015 that gives guidelines for creating and modifying classes/class options. This is a good set of general concepts to use to help with balancing a custom class.
Now that that's out of the way, let's state the obvious...
This is a homebrew character. You cannot do this with the rules as written or as intended. This means working with your DM to create the characters, and only initial analysis and extensive playtesting will ever make a fringe character concept like this balanced.
That said, let's tackle the approach to solving the issue:
Ultimately, the host creature doesn't matter to the parasite/symbiote because only the symbiote needs the powers. The host can simply be a large creature used to hauling around a small one. This means there's really three things you can homebrew to make a character like this work. You can create a custom race, class, or background. It reminds me a bit of Master and Blaster from Mad Max Beyond Thunderdome.
Background
As backgrounds consist only of roughly 4 total proficiencies (2 tool/language, 2 skill), you don't have much to tinker with to get what you want without totally throwing it out of whack, balance wise. I wouldn't look at this as a viable source of the homebrew, but you may want to customize a background to express the symbiotic relationship between the two characters. Riding (Land) and Arcana come to mind with the concept of a 'blood mage' that rides a large creature. Outside of this, refine your concept enough to decide on the other two proficiencies and what package might be appropriate.
Class
This would be closest to what your intent seemed to be, but is also the most work. I actually recommend using this path if your DM is used to creating custom content or willing to put a lot of time in to your character, as balancing a class, even just for class abilities, is arduous work at best.
If you really want to take it down this path, I would probably look at a sorcerer as inspiration for how to tackle the class. It would be a decent archetype for a sorcerer, and fits the overall flavor. It could also be balanced easier. With a sorcerer, you could use the sorcery points to model his ability to draw life from his allies on a point-based basis, and replace the archetype abilities of the sorcerer for appropriate abilities.
Focus on the symbiotic pair abilities first, and the ability to draw from others and feed the 'host' second. Use existing spells to give you an idea of how to approach this.
Race
If you did this as a race, you could limit the capabilities to only cover exactly what you are looking for.
Look at the Teifling (PHB) and the Aasimar homebrew example (DMG) to figure out how to approach this path. You will still need to effectively create new 'spells' with this path in order to create this concept as a race. The biggest problem with this path, however, is that using a race based model, your ability to drain characters other than the host are going to be very limited per day. I would tie the 'first level' spell slot to a flat racial ability to tie your hit points to another creature, just forgoing that slot completely.
Creation and Playtesting
Whether you choose the race or class option to create this character, ensure that you keep in mind that your initial goal. Each ability can fit within a variety of classifications:
One to One Relationship
These abilities can only be shared with a single target... the 'host' of the symbiotic relationship. This means they can likely be just a bit more powerful than the rest of the abilities due to the limitation.
Willing Target
These types of abilities will automatically fail if the target is hostile. This is a good way to keep the character from making unexpected choices in the game that could turn a simple leeching 'spell' in to an offensive 'spell' for attack (like Vampiric Touch). Some of these are likely to be like Cleric spells while others will be more like Wizard spells. Your ideal will likely be that each ability is somewhere in between.
Mixed Target
These abilities would be designed to use on both willing and unwilling targets. These will be the toughest abilities to ensure they are balanced, as they can be used diversely across a wide spectrum of circumstances in ways that are hard to predict. These could be based off of Wizard/Sorcerer necromantic spells easily enough.
Offensive Targets Only
This has the least justification, but should be roughly at the same range of power of most similar Mixed Target spells. This is basically an ability that everyone would innately get their save against, or just have to roll to hit.
Power Range
Initial abilities would likely fall in to the One to One category above, and the symbiote/host relationship is obviously defined here. You can either share hit point totals as a severe weakness to help balance things out, or look to leech hit points from the host. If the latter, then focus the initial abilities on a two for one hit point ratio, or make them cost an expendable resource that is not easily replenished. Once per short rest, or twice per long rest is likely a good place to start. If you went with a custom sorcerer, using Sorcery points is likely what you will want to do, but this means that the relationship does not truly become what you want until 2nd level.
All abilities will end up going through a trial and error/beta-testing phase. Expect abilities to change a bit until they are refined. Changing the targeting type is the easiest way to find the right fit, but don't be afraid to adjust numbers (likely lowering them) of any mechanical effects.
Balance with other classes is important depending on a couple factors. It can't break the fun for other players or yourself, so it can't be far too powerful or far too weak, however, if no other characters (player or non-player) are going to be using this class, it does provide a bit more leniency as far as balancing goes. If they are common in the game world, however, it will take a lot more care to get things just right.
Summation
It is doable, but as you can see from this rather long and extensive answer it is a lot of work. I like the idea of a blood mage of sorts, and feel it is a natural fit for a sorcerer, but ymmv.
Don't expect this concept to fly unless your DM has already expressed a willingness to do the extra work for the class. It's difficult to balance, and touchy ground overall in trying to get everything just right. Expect your character to change frequently, especially early on.
That said, if your DM is good with the idea and willing to work, have fun with it.