hmmm, I think the concept is pretty interesting, but I don't think the kit is too well suited to what you want it to do because you wouldn't really benefit from getting in melee range outside of the ult. Anyways, my thoughts on the kit:
Passive: I think the effect is fine, but I feel the implementation could be a bit better. For instance, why not make it so that the passive applies after X amount of auto attacks and have it do a flat amount of magic damage to the target hit and 2 other targets. I feel this would be a bit better because it would encourage you to stay in melee range to increase your overall dps. What you have now is kinda like J4's passive where you can essentially run up and nuke the enemy in lane every 6 seconds.
Q: I'm a tad confused about this spell. Does the splash damage apply to the targeted enemy? or only to the enemies surrounding the enemy. Anyways, I think this skill is fine, although the numbers could be tweaked a bit. I think the spell should have a little more base damage if the splash doesn't apply to the target because I feel this skill is your most consistant source of damage. If the splash damage does apply to the targeted enemy, you'll have to decrease the base damage of this spell because 300 base damage on a champ with essentially 4 damaging abilities is a bit too much because most champs have a base damage of 250 when they max a skill. The only champ I can think of who breaks this convention is Zilean and he only has 1 damaging skill. Also, most single target nukes have a scaling of .6 AP, so you should probably adjust the scaling too.
W: Again, I think this skill is fine, but it's numbers should be tweaked a bit. Specifically, I think the snare duration should be reduced by .5 at all levels because 2.5 seconds at max rank is a pretty long time, especially compared to some other champs (i.e. maokai, snare duration 2 seconds at max). The only other champs who have hard CC that I can think of that is longer than 2.5 seconds is rammus and morgana and these long duration CC is fine on them because they both have relatively long cool downs and they don't have 4 damaging skills.
E: I have quite a few problems with skill. First of all, what does the "shield strength" refer to? does she gain a shield when this is cast or something? Anyways, the numbering on this skill is kinda all over the place in that the slow is too little to actually keep the enemy within the spell's radius till you're able to reactivate the spell, the damage is way too high for a basic AOE spell. The only thing I can compare this skill to is nunu's ult except it does a lot less damage and it's not very good at keeping the enemies inside the spell's radius. What I suggest you do is that you should either have the slow scale with levels and decrease the damage of the spell and make it a bit easier to pop on enemies or you could change the spell completely. If you decide to change the spell I would recommend making the spell so that it allows the champ to mitigate damage or some sort of sustain because a spell like that would make it easier for this champ to survive in team fights.
R: My feelings towards the ult right now are mixed because it doesn't really mesh that well with the kit so far. What I see is that you have this great damage steroid on this champ who has a hard time getting in or out of battle and also has a hard time living through the battle. So I feel that this ult is pretty bad on the current kit. Also, you dodn't really seem to benefit from the rage explosion because you can't control when the explosion goes off.
In summary, the skills are pretty straight forward, but they don't really mesh that well because half the kit is designed for a mage and half the kit is designed for a melee character. What this devision will cause is that the champ will either be a really good mage or a really good fighter (I'll call this the Sion effect), but a bad melee Mage. What you should do is remove some of the damage and add in a mobility spell or some kind of spell that will make the character more tanky so she can actually survive in melee range or be able to unload her skills on a champ and then get out. Also, I suggest you should look at either Rumble's, Nidalee's, Jax's, or Sion's kit to help you find what you're striving to do in this kit.
I hope this helps.