Ok I know this sounds like a silly question, but can you use charge and Tk together?

Charge says go half and then make a close combat attack
and Tk says do a close combat attack...

They can not be used together. First off, let's be sure you're looking at the Indy Power Card. Secondly, Charge states:
"CHARGE (optional): Give this character a power action. Move this character up to half its speed value and then make one close combat action as a free action. While this power is showing on its dial, this character is not affected by Force Blast and ignores knockback."

While Telekinesis states:
"TELEKINESIS (optional): Give this character a power action and choose one of the following:..."

Since Charge allows for a close combat action but TK requires a power action, they can not be used together.

who has this combination of powers?

I believe the new LEs for the DC set have them (Hal Jordan and Maxima Whateverhernameis). Other than that, not sure.

You only actually do a close combat attack when TKing an opposing figure. When moving a friendly figure, it is considered a power action, so yes, as far as I know, you can use charge and TK.

Never mind, I thought you meant can you TK a figure out and have it charge an opposing figure. Oops