What is the Buddhist perception of injustice, considering the law of Karma? Considering the notion of our future being the result of our actions in the past, is there any room for the concept of unfairness in Buddhist philosophy? What is the right/skillful view one should take when witnessing suffering in a loved one? 

From my understanding, one should be compassionate towards the person and keep a calm mind as much as possible, but at the same time I find discomforting the idea that somehow, the present condition of a loved one has to do with his/her past actions, specially when considering more extreme examples of suffering without any apparent reason (e.g. childhood traumatic events, such as sexual abuse).
It's certainly been seen as a difficult question for Buddhists. I've heard it termed the Glenn Hoddle question after an ex-footballer used karma type arguments to explain disability - to quote


  You and I have been physically given two hands and two legs and
  half-decent brains. Some  people have not been born like that for a
  reason. The karma is working from another lifetime


He was sacked as England coach. However it is a difficult question and I will attempt a decent answer

Karma isn't cause and effect

Karma isn't just cause and effect - there is more going on. To quote the Sivaka Sutra


  Now when these ascetics and brahmans have such a doctrine and view
  that 'whatever a person experiences, be it pleasure, pain or
  neither-pain-nor-pleasure, all that is caused by previous action,'
  then they go beyond what they know by themselves and what is accepted
  as true by the world. Therefore, I say that this is wrong on the part
  of these ascetics and brahmans.


It's not as if someone lies in one life then is reborn as with a disability in the next. Sometimes the fruits of karma (action) are immediate, sometimes they are delayed, sometimes the fruits don't come to pass at all. 

There is more going on than Karma

Also in Buddhism there is the concept of Niyamas which explains outcome dependant origination (the underpinning of reality). These being


kamma niyama: ("action") consequences of one's actions
utu niyama: ("time, season") seasonal changes and climate, law of non-living matter
bīja niyama: ("seed") laws of heredity
citta niyama:("mind") will of mind
dhamma niyama: ("law") nature's tendency to perfect


The above list is taken from here. So karma would only explain a fraction of someones present conditions.

Karma isn't blame

I guess the other thing to point out is that there is no concept of blame in any of this. The audio here is very interesting - a Buddhist nun discusses this exact issue with someone with seeing difficulties. 

Even if someone's unfortunate circumstances were caused directly by their actions (which is no means as simple as that) then this wouldn't stop us feeling compassion for them. This would be true in this life or the next. If someone was run over because they didn't look both ways before crossing the road then it is a normal human response to feel compassion for them - Buddhist or not.

As I say though a difficult and interesting question. I hope that helps some.