Change position of "proceed to checkout" button

It's on the right by default.
If you don't see it on the right, it means that your template's CSS overrides the CSS in HikaShop. So you should edit the hikashop CSS and change the code:
#hikashop_checkout_next_button{
float:right;
}

to:
#hikashop_checkout_next_button{
float:right !important;
}

Yes. The button as the class hikashop_cart_input_button and the id hikashop_checkout_next_button

You should have the id after the value. If you don't, maybe it's because you edited the file step of the view checkout where the id parameter is given to the next button:
echo $this->cart->displayButton(JText::_('HIKA_NEXT'),'next',$this->params,hikashop_completeLink('checkout&task=step&step='.($this->step+1)),'if(hikashopCheckChangeForm(\'order\',\'hikashop_checkout_form\')){document.getElementById(\'hikashop_validate\').value=1; document.forms[\'hikashop_checkout_form\'].submit();} return false;','id="hikashop_checkout_next_button"');

Wow sorry. I was thinking about the cart view of the checkout.
Indeed, that's not the good thing to do for what you want

You should try to add the CSS:
#hikashop_cart_module .button hikashop_cart_input_button{
float: right !important;
}