How do Christians justify their belief and how do they convince others of their belief?

I could tell you how I justify my belief (and it is based on logic, which is based on assumptions), but it wouldn't provide a universal answer worthy of a StackExchange site. What makes sense to me may resonate with a lot of Christians, or very few, or none.

In all honesty, I probably sound like a fool to the majority of people, yet to me, my reason for believing is as solid as the belief itself.

The answer to this question is unique to every Christian. We can't give an answer that applies universally. Not only are we all individuals, with different ways of viewing the world around us, we are also all called by God in a specific way, unique to each of us.

Not everything that is true can be logically proven. Think about interpersonal relationships and your feelings for another person. If you contend that all of your feelings for other people can be based on or explained by the laws of logic, I would assume you've never been in love.

I can't really speak for other traditions, but the Evangelical Protestant view is that Christianity is a relationship, not a religion. (I can't really speak for other traditions.) Christians are those who have responded to God's invitation to a personal relationship with Him. Like human relationships, there are aspects of this that cannot be logically proven, yet are true.

Lastly, I assume that Christians respect the laws of logic and therefore their justifications must also follow these laws.

Then you're not really going to get very far. Yes, Christians respect the laws of logic, but we do not respect only the laws of logic. The basis of Christian belief isn't grounded in logic, but in testimony, faith and experience. We know that God is there because he makes his presence known through the witness of the Holy Ghost, and we know that he is good and worthy of obedience and worship because of the history of our relationship with him.

As we follow God's commandments faithfully, we can see the results in our lives. We see that living the principles of the Gospel works, that it produces positive results, helping us to resolve or avoid problems in our lives. That increases our faith, making us better-able to trust in God, so that when we need to do something truly difficult, even if it seems to contradict what we think is logical, we can say, "God has never led me wrong before, so I can trust in him now." Those who don't understand this principle say that Christians live by "blind faith," but that is far from the truth.

As for how Christians convince others of their belief, we don't. While it's possible to convince another person that our viewpoint makes sense, actual faith, the kind I'm talking about in the previous paragraphs, comes from God, not from another person. What proselyting Christians attempt to do is convince others to open their hearts, to be willing to accept that the Gospel might be true, and to be willing to pray to God to seek their own testimony and establish their own relationship with him. Conversion is a highly personal matter between oneself and the Lord; anyone else's involvement is secondary.

The formulation of this question is broad as has been noted, but I will answer this as best I can.

We justify our beliefs based on a series of foundational beliefs that require faith, but then have logical outcomes and structures that follow. The book of Romans does an excellent job laying out these basics and their implications. As such, I have found that you can logically justify the Christian faith and its beliefs, as long as you have faith and believe (accept as factually true) the basic foundational beliefs. If you believe there is a God, and if you believe that the Bible is what it says it is then you can logically form (a VERY long) argument for all facets of the Christian faith.

However, this would still just be head knowledge. Let's forget labels for just one second as they tend confuse things. To be a follower of Christ, to be "saved" or within His elect, you must believe that Christ was real, historical, bodily resurrected and alive today, just not with us here on this earth. You also must love and trust Jesus to fulfill His promises and show you grace and mercy. This is the essence of being a Christian, and in a very real way I cannot justify or prove this to you anymore than I can "prove" why I love my fiancee.