The answer to this question, based on Scripture, is somewhat speculative. But there are a few passages which seem to suggest that our deceased relatives probably can see us. Let's consider, for example, Hebrews 12...
Therefore, since we are surrounded by so great a cloud of witnesses, let us also lay aside every weight, and sin which clings so closely, and let us run with endurance the race that is set before us, looking to Jesus, the founder and perfecter of our faith, who for the joy that was set before him endured the cross, despising the shame, and is seated at the right hand of the throne of God. (Hebrews 12:1-2)
This passage refers to our being “surrounded by so great a cloud of witnesses”. Where it says “let us run with endurance the race that is set before us...” it is an image of our lives being like a race, and the “cloud of witnesses” are those watching us in the stands. Who are these people? There seem to be a few options. It could be the unbelievers in the world who watch us and see the truth of the gospel from the way we live our lives. Or it could be the angels in heaven who watch us (1 Corinthians 4:9). But the previous chapter of Hebrews spoke about all of the men and women of faith in the Bible, and it served to encourage us in our faith. It ends by telling us that they have not yet received the goal of their faith, as God will perfect us all together. And so it seems that this passage, which follows directly on, would refer to those people. By this passage, it does seem that they are able to watch the activity of the Church in the world.
I suppose that we might also consider the rich man who died in Luke 16, and was concerned for the salvation of his family on Earth. Or we might consider how the Bible speaks of mediums, affirming that we can summon the dead. Samuel was summoned from the dead and seemed to know about the current state of affairs, and predicted the imminent death of Saul. Other stronger evidences, if there are any, don't come to mind right now; and all of these are not entirely conclusive. But if our loved ones can see us, how should this affect our lives, if at all? In Hebrews 12, it seems to suggest that this fact should affect our lives. The writer of Hebrews is saying “since we are surrounded by witnesses, let us also lay aside every weight, and sin...” But we live for Christ, and not to please men; least of all those who are no longer with us. And this passage above still urges us to be “looking to Jesus, the founder and perfecter of our faith...” We must not try to communicate with the dead; consulting mediums is strictly forbidden by God. And we should not pray to the dead, but we pray directly to Christ. “For there is one God, and there is one mediator between God and men, the man Christ Jesus.” (1 Timothy 2:5)
I was talking to someone a while ago, and she told me that she's been keeping a “spiritual journal”. I told her how wonderful I thought that was, and I also happened to say something like “That journal will be a great blessing to your children when you're gone.” Well, she wasn't too keen on that. She didn't like the idea that anyone would ever read her journal; not even family. This got me thinking. Jesus said, “For everyone who does wicked things hates the light and does not come to the light, lest his deeds should be exposed.” (John 3:20). Now it's probably not that this woman is writing horrible things about people all the time, but I'm wondering what one might write in their journal that they would be ashamed of if their family were to read it after they’re dead? There should be no such things written there. And I thought of those I know who have died... What am I doing that I ought to be ashamed of before them? The sins I kept hidden from them when they were alive may be fully exposed to them now!
But that they can see me, or that my family might read my journal when I'm dead... this should not be my concern. Rather, Jesus right now sees all that I do, and what I write in my journal, and even what I think in my head. I should be far more concerned about pleasing Him than anyone else, dead or alive.
Now we all sin, and we are all ashamed of our sins. We know that God is always watching, but because of who He is, we can accept that. He knows our weaknesses, and He loves us anyway. As for our deceased loved ones; they would now have a very different perspective on the world. From where they are now, they understand the degree to which the whole world is corrupted by sin, and how they themselves were also. I imagine they would be far more sympathetic towards us. But one day, when we are all perfected in Christ and reunited with one another, love will cover a multitude of sins! But in the present, our lives should be holy before Christ first, others second; and we can long for the day when we are reunited with those who have passed.
I leave you with this...
Only let your manner of life be worthy of the gospel of Christ, so that whether I come and see you or am absent, I may hear of you that you are standing firm in one spirit, with one mind striving side by side for the faith of the gospel (Philippians 1:27)