Postdoc application, cold contact, funding proposal Backgrounds: I’m an international researcher (biologist) in Europe; with a PhD and one year postdoctoral experience. I want to increase my chances to get a second postdoctoral position through contacting potential PIs and expressing my willingness to apply for my own funding and to write a postdoctoral research grant applications. 

Questions:

1)  When I cold contact professors for postdoc positions, what is the most polite way; Should I immediately (with my cv) send a cover letter including research ideas and informing them that I want to apply for my own grant? Or just present myself and my research interests and wait for a reply. Then if it is a yes I fellow up with my ideas etc..

2)   Often Professors are busy and have big research group to handle so I feel like they might not be interested in hassling for a grant with a new postdoc. Especially if they don’t know him or his PhD advisor, so they tend to not reply. (I don’t blame them though).
So is it better to contact the second most senior researcher in the team, to improve the chances to get a reply?
I think that you are throwing in too many variables here. Your plan might work, but it depends a lot on who you contact so a firm answer here is probably out of the question. 

The most important thing to keep in mind is that you need to convince them that they want you and your skills. Everything else is secondary. If you first convince them that you have indispensable skills you are in, otherwise they are taking a chance. 

As to your funding/grant proposal. If you had a grant in hand it might be different, but you may be complicating their decision, not simplifying it. If the grant doesn't come through where are you. More important from their standpoint is "Where are we?". 

On the other hand if you are already a successful grant writer as well as a fantastic biologist then it is a point in your favor. 

Cold contacts are always more risky than more naturally developed relationships of course. I don't think your plan changes that very much. Personal contacts through your advisor or current PI are much more likely to meet success. 

On the other hand, if someone expresses a lot of interest in you and has lab/office space available but no funding, you might then bring up the idea of seeking your own funding.