I would like to add my personal perspective which is not too dissimilar from your experience. Some background:
I am a trained wet-lab biologist and used live-cell imaging, molecular biology and cell culture in the context of neurobiology up to my 1st posdoc. Then, due to some results during my PhD, I decided to make a change to study regulation of gene expression in a cell biology lab. The project was mostly wet-lab, but it already had the promise of large datasets and the opportunity to learn how to analyse that data. For a number of reasons, at some point I was mostly sitting on the computer mapping NGS data and plotting in R. And I liked it a lot. So much so, that my supervisor supported me in taking some courses and leaving the wet-lab for others while I did the analysis. Fast forward a couple of years and now I do bioinformatics.
Bioinformatics or computational biology? Even though there is not established distinction between what constitutes one or the other, in my mind one is more about data analysis and the other more about developing algorithms. The latter has stronger CS component in my view, and the two often go hand-in-hand of course. So do you want to develop algorithms or are you more interested in teasing out answers from data asap? Because, IMO, the path is different. Also bear in mind that bioinformatics and computational biology are very broad churches. You could be doing analysis NGS data from -seq, doing some protein network analysis, running simulations of evolution, or even developing algorithms for data extraction from images. Even though Biostar focus more on the gen side, there are many more things out there.
Now, I will answer your questions assuming that you are more interested in bioinformatics.
How can I make that switch after my PhD?
For your post-doc choose a lab with a project that you like (very important), and that has both components (wet+dry), or is looking for someone willing to learn the bioinformatics bit. There are more projects/labs like that than you think, specially if you are willing to work with NGS. It is important that if you are going to learn "on the job", as a lot of us do, there will be people next to you to help you in the beginning. This could be a lab colleague or a collaborator. Failing that, the lab should allow you take the courses needed to learn.
In the meantime, you can start by taking a few courses even during your PhD to learn the basics. Let's say how to work with the command-line or programming in phyton/perl/java and R. Your university might have some courses available that you can take, or you can attend a (free) software-carpentry workshop. Learning how to use the command-line is extremely important and useful regardless of what you do. There are also two books that I earthly recommend (getting one of them his enough though - unless there is grant money available):
Both are written from a very practical perspective and will get you started almost immediately. The 1st also contains chapters on python, and the second spends more time evangelizing on the the power of reproducible research. They sort of complement each other, but the first is better to get things done from page 1 and the second is better if you are serious about bioinformatics in the long run. Both will teach you how to use the command-line and it will save you time as soon as you start!
If you don't use Linux or a MacOS, set-up a computer with Linux (Ubuntu*) dual-boot or a virtualBox. Also pay more attention to statistics. Take a course at Uni if possible, or do one online.
Or is it even possible to integrate some computational tools in my current research? (I have no idea).
Without knowing exactly what your project is I can't really answer, but chances are that you can. For e.g. during my PhD, which involved cloning receptors, I had to do homology searches, domain predictions, and analysis of imaging data. I did not knew it at the time, but learning some bioinformatics (and R) would have helped a lot.
What can I do now to find a post-doc position or research scientist position in computational biology after my phd?
To be honest,for you to get a completely dry lab position after your PhD you would have to learn quite a bit in the meantime. That said, if you can learn enough to write a chapter your thesis from analysis that you performed, it will go a long way. Remember that most labs/positions will know that a pos-doc does not know everything, but brings something. So say, you go to a computational lab that works in your current field, but from a different perspective, you will bring them biological know-how, and in turn they will teach you the computational side specially if you already shown an active interest in learning.
Should I take computational biology certificate course after my PhD?
When I was planning for the switch I considered taking a step back and doing an MSc in bioinformatics. At the time I asked a few bioinformaticians that knew what I was doing and they suggested it was a bit of a waste of time. They believed that an MSc would not teach me enough to be worth the time - besides I am more interested in answering question than developing algorithms/software. In subsequent job applications/interviews a formal education in CB was not never an issue. Showing that I could do the analysis was more relevant. That said, depending on what your goal is, some formal education might be worth it. If I could go back I would have probably taken a class in CS and statistics while doing my PhD.
Given the lack of detail in your post it hard to give more specific advice, but I hope this helps. If there something I learned from working with people that came to science at different stages, and changed fields relatively late in the career, is that it is never too late.
*other flavours are available
[edited for grammar]
What exactly is your line of bench work and what methods do you use?